{"history":[{"id":"e4652df9-f4a8-457a-a027-b717a7fa77f1","wordChanged":"**Taylor series**","newWord":":::summary\n:::\n\n# \n[[File:sintay_SVG.svg|thumb|300px|As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows\nsin x\nand its Taylor approximations by polynomials of degree **1**, **3** , **5** , **7** , **9** , **1****1** , and **1****3** at\nx = 0\n.]]\nIn [mathematics](https://bliptext.com/articles/mathematics), the Swift or **Taylor expansion** of a [function](https://bliptext.com/articles/function-mathematics) is an [infinite sum](https://bliptext.com/articles/series-mathematics) of terms that are expressed in terms of the [function](https://bliptext.com/articles/function-mathematics)'s [derivatives](https://bliptext.com/articles/derivative) at a single point.\nFor most common functions, the function and the sum of its Taylor series are equal near this point.\nTaylor series are named after [Brook Taylor](https://bliptext.com/articles/brook-taylor), who introduced them in 1715.\nA Taylor series is also called a **Maclaurin series** when 0 is the point where the derivatives are considered, after [Colin Maclaurin](https://bliptext.com/articles/colin-maclaurin), who made extensive use of this special case of Taylor series in the 18th century.\nThe [partial sum](https://bliptext.com/articles/partial-sum) formed by the first\nn + 1\nterms of a Taylor series is a [polynomial](https://bliptext.com/articles/polynomial) of degree n that is called the nth **Taylor [polynomial](https://bliptext.com/articles/polynomial)** of the function.\nTaylor polynomials are approximations of a function, which become generally more accurate as n increases.\n[Taylor's theorem](https://bliptext.com/articles/taylor-s-theorem) gives quantitative estimates on the error introduced by the use of such approximations.\nIf the Taylor series of a function is [convergent](https://bliptext.com/articles/convergence-mathematics), its sum is the [limit](https://bliptext.com/articles/limit-of-a-sequence) of the [infinite sequence](https://bliptext.com/articles/infinite-sequence) of the Taylor polynomials.\nA function may differ from the sum of its Taylor series, even if its Taylor series is convergent.\nA function is [analytic](https://bliptext.com/articles/analytic-function) at a point x if it is equal to the sum of its Taylor series in some [open interval](https://bliptext.com/articles/open-interval) (or [open disk](https://bliptext.com/articles/disk-mathematics) in the [complex plane](https://bliptext.com/articles/complex-plane)) containing x. This implies that the function is [analytic](https://bliptext.com/articles/analytic-function) at every point of the interval (or disk).\n\n# Definition\nThe Taylor series of a [real](https://bliptext.com/articles/real-valued-function) or [complex-valued function](https://bliptext.com/articles/complex-valued-function)\nf (x)\n, that is [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable-function) at a [real](https://bliptext.com/articles/real-number) or [complex number](https://bliptext.com/articles/complex-number)\na\n, is the [power series](https://bliptext.com/articles/power-series)\nHere,\nn!\ndenotes the [factorial](https://bliptext.com/articles/factorial) of n. The function\nf(a)\ndenotes the nth [derivative](https://bliptext.com/articles/derivative) of f evaluated at the point a. The [derivative](https://bliptext.com/articles/derivative) of order zero of f is defined to be f itself and\n(x − a)0\nand\n0!\n[are both defined to be 1](https://bliptext.com/articles/empty-product).\nThis series can be written by using [sigma notation](https://bliptext.com/articles/sigma-notation), as in the right side formula.\nWith\n, the Maclaurin series takes the form:\n\n# Examples\nThe Taylor series of any [polynomial](https://bliptext.com/articles/polynomial) is the [polynomial](https://bliptext.com/articles/polynomial) itself.\nThe Maclaurin series of\n1⁄1 − x\nis the [geometric series](https://bliptext.com/articles/geometric-series)\nSo, by substituting x for\n1 − x\n, the Taylor series of\n1⁄x\nat\nis\nBy integrating the above Maclaurin series, we find the Maclaurin series of\nln(1 − x)\n, where\nln\ndenotes the [natural logarithm](https://bliptext.com/articles/natural-logarithm):\nThe corresponding Taylor series of\nln x\nat\nis\nand more generally, the corresponding Taylor series of\nln x\nat an arbitrary nonzero point a is:\nThe Maclaurin series of the [exponential function](https://bliptext.com/articles/exponential-function)\nex\nis\nThe above expansion holds because the derivative of\nex\nwith respect to x is also\nex\n, and\ne0\nequals 1.\nThis leaves the terms\n(x − 0)n\nin the numerator and\nn!\nin the denominator of each term in the infinite sum.\n\n# History\nThe [ancient Greek philosopher](https://bliptext.com/articles/ancient-greek-philosopher) [Zeno of Elea](https://bliptext.com/articles/zeno-of-elea) considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility; the result was [Zeno's paradox](https://bliptext.com/articles/zeno-s-paradox).\nLater, [Aristotle](https://bliptext.com/articles/aristotle) proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by [Archimedes](https://bliptext.com/articles/archimedes), as it had been prior to [Aristotle](https://bliptext.com/articles/aristotle) by the Presocratic Atomist [Democritus](https://bliptext.com/articles/democritus).\nIt was through Archimedes's [method of exhaustion](https://bliptext.com/articles/method-of-exhaustion) that an infinite number of progressive subdivisions could be performed to achieve a finite result.\n[Liu Hui](https://bliptext.com/articles/liu-hui) independently employed a similar method a few centuries later.\nIn the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by Indian mathematician [Madhava of Sangamagrama](https://bliptext.com/articles/madhava-of-sangamagrama).\nThough no record of his work survives, writings of his followers in the [Kerala school of astronomy and mathematics](https://bliptext.com/articles/kerala-school-of-astronomy-and-mathematics) suggest that he found the Taylor series for the [trigonometric functions](https://bliptext.com/articles/trigonometric-function) of [sine](https://bliptext.com/articles/sine), co[sine](https://bliptext.com/articles/sine), and [arctangent](https://bliptext.com/articles/arctangent) (see [Madhava series](https://bliptext.com/articles/madhava-series)).\nDuring the following two centuries his followers developed further series expansions and rational approximations.\nIn late 1670, [James Gregory](https://bliptext.com/articles/james-gregory-mathematician) was shown in a letter from [John Collins](https://bliptext.com/articles/john-collins-mathematician) several Maclaurin series (\\sin x, \\cos x, \\arcsin x, and x \\cot x) derived by [Isaac Newton](https://bliptext.com/articles/isaac-newton), and told that Newton had developed a general method for expanding functions in series.\nNewton had in fact used a cumbersome method involving long division of series and term-by-term integration, but Gregory did not know it and set out to discover a general method for himself.\nIn early 1671 Gregory discovered something like the general Maclaurin series and sent a letter to Collins including series for \\arctan x, \\tan x, \\sec x, (the integral of \\tan), (the [[integral of the secant function|integral of\nsec\n]], the inverse [Gudermannian function](https://bliptext.com/articles/gudermannian-function)), and (the [Gudermannian function](https://bliptext.com/articles/gudermannian-function)). However, thinking that he had merely redeveloped a method by Newton, Gregory never described how he obtained these series, and it can only be inferred that he understood the general method by examining scratch work he had scribbled on the back of another letter from 1671.\nIn 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work De Quadratura Curvarum.\nHowever, this work was never completed and the relevant sections were omitted from the portions published in 1704 under the title Tractatus de Quadratura Curvarum.\nIt was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by [Brook Taylor](https://bliptext.com/articles/brook-taylor), after whom the series are now named.\nThe Maclaurin series was named after [Colin Maclaurin](https://bliptext.com/articles/colin-maclaurin), a Scottish mathematician, who published the special case of the Taylor result in the mid-18th century.\n\n# Analytic functions\n[[Image:Exp neg inverse square.svg|upright=1.4|thumb|right|The function\ne(−1/x 2)\nis not analytic at\nx = 0\nIf\nf (x)\nis given by a convergent power series in an open disk centred at b in the complex plane (or an interval in the real line), it is said to be [analytic](https://bliptext.com/articles/analytic-function) in this region.\nThus for x in this region, f is given by a convergent power series\nDifferentiating by x the above formula n times, then setting\ngives:\nand so the power series expansion agrees with the Taylor series.\nThus a function is analytic in an open disk centered at b if and only if its Taylor series converges to the value of the function at each point of the disk.\nIf\nf (x)\nis equal to the sum of its Taylor series for all x in the complex plane, it is called [entire](https://bliptext.com/articles/entire-function).\nThe polynomials, [exponential function](https://bliptext.com/articles/exponential-function)\nex\n, and the [trigonometric functions](https://bliptext.com/articles/trigonometric-function) sine and cosine, are examples of entire functions.\nExamples of functions that are not entire include the [square root](https://bliptext.com/articles/square-root), the [logarithm](https://bliptext.com/articles/logarithm), the [trigonometric function](https://bliptext.com/articles/trigonometric-function) tangent, and its inverse, [arctan](https://bliptext.com/articles/arctan).\nFor these functions the Taylor series do not [converge](https://bliptext.com/articles/convergent-series) if x is far from b. That is, the Taylor series [diverges](https://bliptext.com/articles/divergent-series) at x if the distance between x and b is larger than the radius of [converge](https://bliptext.com/articles/convergent-series)nce.\nThe Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point.\nUses of the Taylor series for analytic functions include:\n\n# Approximation error and convergence\n[[Image:LogTay.svg|upright=1.4|thumb|right|The Taylor polynomials for\nln(1 + x)\nonly provide accurate approximations in the range\n−1 < x ≤ 1\n. For\nx > 1\n, Taylor polynomials of higher degree provide worse approximations.]]\n[[Image:Logarithm GIF.gif|upright=1.4|thumb|right|The Taylor approximations for\nln(1 + x)\n(black).\nFor\nx > 1\n, the approximations diverge.]]\nPictured is an accurate approximation of\nsin x\naround the point\n. The pink curve is a polynomial of degree seven:\nThe error in this approximation is no more than\n. For a full cycle centered at the origin (\n−π < x < π\n) the error is less than 0.08215. In particular, for\n−1 < x < 1\n, the error is less than 0.000003.\nIn contrast, also shown is a picture of the natural logarithm function\nln(1 + x)\nand some of its Taylor polynomials around\n. These approximations converge to the function only in the region\n−1 < x ≤ 1\nThe error incurred in approximating a function by its nth-degree Taylor polynomial is called the remainder or [residual](https://bliptext.com/articles/residual-numerical-analysis) and is denoted by the function\nRn(x)\n. Taylor's theorem can be used to obtain a bound on the [size of the remainder](https://bliptext.com/articles/taylor-s-theorem).\nIn general, Taylor series need not be [convergent](https://bliptext.com/articles/convergent-series) at all.\nIn fact, the set of functions with a convergent Taylor series is a [meager set](https://bliptext.com/articles/meager-set) in the [Fréchet space](https://bliptext.com/articles/fr-chet-space) of [smooth functions](https://bliptext.com/articles/smooth-functions).\nEven if the Taylor series of a function f does converge, its limit need not be equal to the value of the function\nf (x)\n. For example, the function\nis [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable) at\n, and has all derivatives zero there.\nConsequently, the Taylor series of\nf (x)\nabout\nis identically zero.\nHowever,\nf (x)\nis not the zero function, so does not equal its Taylor series around the origin.\nThus,\nf (x)\nis an example of a [non-analytic smooth function](https://bliptext.com/articles/non-analytic-smooth-function).\nIn [real analysis](https://bliptext.com/articles/real-analysis), this example shows that there are [infinitely differentiable functions](https://bliptext.com/articles/infinitely-differentiable-function)\nf (x)\nwhose Taylor series are not equal to\nf (x)\neven if they converge.\nBy contrast, the [holomorphic functions](https://bliptext.com/articles/holomorphic-function) studied in [complex analysis](https://bliptext.com/articles/complex-analysis) always possess a convergent Taylor series, and even the Taylor series of [meromorphic functions](https://bliptext.com/articles/meromorphic-function), which might have singularities, never converge to a value different from the function itself.\nThe complex function\ne−1/z 2\n, however, does not approach 0 when z approaches 0 along the imaginary axis, so it is not [continuous](https://bliptext.com/articles/continuous-function) in the complex plane and its Taylor series is undefined at 0.\nMore generally, every sequence of real or complex numbers can appear as [coefficients](https://bliptext.com/articles/coefficient) in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of [Borel's lemma](https://bliptext.com/articles/borel-s-lemma).\nAs a result, the [radius of convergence](https://bliptext.com/articles/radius-of-convergence) of a Taylor series can be zero.\nThere are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere.\nA function cannot be written as a Taylor series centred at a [singularity](https://bliptext.com/articles/singularity-mathematics); in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see [Laurent series](https://bliptext.com/articles/laurent-series).\nFor example,\nf (x) = e−1/x 2\ncan be written as a Laurent series.\n# [The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin. | upload.wikimedia.org/wikipedia/commons/b/b6/Taylorsine.svg]\n\n## Generalization\nThe generalization of the Taylor series does converge to the value of the function itself for any [bounded](https://bliptext.com/articles/bounded-function) [continuous function](https://bliptext.com/articles/continuous-function) on\n(0,∞)\n, and this can be done by using the calculus of [finite differences](https://bliptext.com/articles/finite-differences).\nSpecifically, the following theorem, due to [Einar Hille](https://bliptext.com/articles/einar-hille), that for any\nt > 0\n,\nHere\nΔn h\nis the nth finite difference operator with step size h. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the [Newton series](https://bliptext.com/articles/newton-series).\nWhen the function f is analytic at a, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series.\nIn general, for any infinite sequence\nai\n, the following power series identity holds:\nSo in particular,\nThe series on the right is the [expected value](https://bliptext.com/articles/expected-value) of\nf (a + X)\n, where X is a [Poisson-distributed](https://bliptext.com/articles/poisson-distribution) [random variable](https://bliptext.com/articles/random-variable) that takes the value\njh\nwith probability\ne−t/h·(t/h)⁄j!\n. Hence,\nThe [law of large numbers](https://bliptext.com/articles/law-of-large-numbers) implies that the identity holds.\n\n# List of Maclaurin series of some common functions\nSeveral important Maclaurin series expansions follow.\nAll these expansions are valid for complex arguments x.\n\n## Exponential function\nExp series.gif\nex\n(in blue), and the sum of the first\nn + 1\nterms of its Taylor series at 0 (in red).]]\nThe [exponential function](https://bliptext.com/articles/exponential-function) e^x (with base e) has Maclaurin series\nIt converges for all x.\nThe exponential [generating function](https://bliptext.com/articles/generating-function) of the [Bell numbers](https://bliptext.com/articles/bell-number) is the exponential function of the predecessor of the exponential function:\n\n## Natural logarithm\nThe [natural logarithm](https://bliptext.com/articles/natural-logarithm) (with base e) has Maclaurin series\nThe last series is known as [Mercator series](https://bliptext.com/articles/mercator-series), named after [Nicholas Mercator](https://bliptext.com/articles/nicholas-mercator) (since it was published in his 1668 treatise Logarithmotechnia).\nBoth of these series converge for |x| < 1.\n(In addition, the series for\nln(1 − x)\nconverges for\n, and the series for\nln(1 + x)\nconverges for\n.)\n\n## Geometric series\nThe [geometric series](https://bliptext.com/articles/geometric-series) and its derivatives have Maclaurin series\nAll are convergent for |x| < 1.\nThese are special cases of the [binomial series](https://bliptext.com/articles/) given in the next section.\n\n## Binomial series\nThe [binomial series](https://bliptext.com/articles/binomial-series) is the power series\nwhose coefficients are the generalized [binomial coefficients](https://bliptext.com/articles/binomial-coefficient)\n(If\nn = 0\n, this product is an [empty product](https://bliptext.com/articles/empty-product) and has value 1.) It converges for |x| < 1 for any real or complex number α.\nWhen\nα = −1\n, this is essentially the infinite geometric series mentioned in the previous section.\nThe special cases\nα = 1⁄2\nand\nα = −1⁄2\ngive the [square root](https://bliptext.com/articles/square-root) function and its [inverse](https://bliptext.com/articles/multiplicative-inverse):\nWhen only the [linear term](https://bliptext.com/articles/linear-approximation) is retained, this simplifies to the [binomial approximation](https://bliptext.com/articles/binomial-approximation).\n\n## Trigonometric functions\nThe usual [trigonometric functions](https://bliptext.com/articles/trigonometric-function) and their inverses have the following Maclaurin series:\nAll angles are expressed in [radians](https://bliptext.com/articles/radian).\nThe numbers\nBk\nappearing in the expansions of\ntan x\nare the [Bernoulli numbers](https://bliptext.com/articles/bernoulli-numbers).\nThe\nEk\nin the expansion of\nsec x\nare [Euler numbers](https://bliptext.com/articles/euler-number).\n\n## Hyperbolic functions\nThe [hyperbolic functions](https://bliptext.com/articles/hyperbolic-function) have Maclaurin series closely related to the series for the corresponding trigonometric functions:\nThe numbers\nBk\nappearing in the series for\ntanh x\nare the [Bernoulli numbers](https://bliptext.com/articles/bernoulli-numbers).\n\n## Polylogarithmic functions\nThe [polylogarithms](https://bliptext.com/articles/polylogarithm) have these defining identities:\nThe [Legendre chi functions](https://bliptext.com/articles/legendre-chi-function) are defined as follows:\nAnd the formulas presented below are called [inverse tangent integrals](https://bliptext.com/articles/inverse-tangent-integral):\nIn [statistical thermodynamics](https://bliptext.com/articles/statistical-mechanics) these formulas are of great importance.\n\n## Elliptic functions\nThe complete [elliptic integrals](https://bliptext.com/articles/elliptic-integral) of first kind K and of second kind E can be defined as follows:\nThe [Jacobi theta functions](https://bliptext.com/articles/theta-function) describe the world of the elliptic modular functions and they have these Taylor series:\nThe regular [partition number sequence](https://bliptext.com/articles/partition-function-number-theory) P(n) has this generating function:\nThe strict partition number sequence Q(n) has that generating function:\n\n# Calculation of Taylor series\nSeveral methods exist for the calculation of Taylor series of a large number of functions.\nOne can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern.\nAlternatively, one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series.\nIn some cases, one can also derive the Taylor series by repeatedly applying [integration by parts](https://bliptext.com/articles/integration-by-parts).\nParticularly convenient is the use of [computer algebra systems](https://bliptext.com/articles/computer-algebra-system) to calculate Taylor series.\n\n## First example\nIn order to compute the 7th degree Maclaurin polynomial for the function\none may first rewrite the function as\nthe composition of two functions and The Taylor series for the natural logarithm is (using [big O notation](https://bliptext.com/articles/big-o-notation))\nand for the cosine function\nThe first several terms from the second series can be substituted into each term of the first series.\nBecause the first term in the second series has degree 2, three terms of the first series suffice to give a 7th-degree polynomial:\nSince the cosine is an [even function](https://bliptext.com/articles/even-function), the coefficients for all the odd powers are zero.\n\n## Second example\nSuppose we want the Taylor series at 0 of the function\nThe Taylor series for the exponential function is\nand the series for cosine is\nAssume the series for their quotient is\nMultiplying both sides by the denominator \\cos x and then expanding it as a series yields\nComparing the coefficients of g(x)\\cos x with the coefficients of e^x,\nThe coefficients c_i of the series for g(x) can thus be computed one at a time, amounting to long division of the series for e^x and \\cos x:\n\n## Third example\nHere we employ a method called \"indirect expansion\" to expand the given function.\nThis method uses the known Taylor expansion of the exponential function.\nIn order to expand\n(1 + x)ex\nas a Taylor series in x, we use the known Taylor series of function\nex\nThus,\n\n# Taylor series as definitions\nClassically, [algebraic functions](https://bliptext.com/articles/algebraic-function) are defined by an algebraic equation, and [transcendental functions](https://bliptext.com/articles/transcendental-function) (including those discussed above) are defined by some property that holds for them, such as a [differential equation](https://bliptext.com/articles/differential-equation).\nFor example, the [exponential function](https://bliptext.com/articles/exponential-function) is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin.\nHowever, one may equally well define an [analytic function](https://bliptext.com/articles/analytic-function) by its Taylor series.\nTaylor series are used to define functions and \"[operators](https://bliptext.com/articles/operator-mathematics)\" in diverse areas of mathematics.\nIn particular, this is true in areas where the classical definitions of functions break down.\nFor example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the [matrix exponential](https://bliptext.com/articles/matrix-exponential) or [matrix logarithm](https://bliptext.com/articles/matrix-logarithm).\nIn other areas, such as formal analysis, it is more convenient to work directly with the [power series](https://bliptext.com/articles/power-series) themselves.\nThus one may define a solution of a differential equation as a power series which, one hopes to prove, is the Taylor series of the desired solution.\n\n# Taylor series in several variables\nThe Taylor series may also be generalized to functions of more than one variable with\nFor example, for a function f(x,y) that depends on two variables, x and y, the Taylor series to second order about the point\n(a, b)\nis\nwhere the subscripts denote the respective [partial derivatives](https://bliptext.com/articles/partial-derivative).\n\n## Second-order Taylor series in several variables\nA second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as\nwhere\nD f (a)\nis the [gradient](https://bliptext.com/articles/gradient) of f evaluated at\nand\nD2 f (a)\nis the [Hessian matrix](https://bliptext.com/articles/hessian-matrix).\nApplying the [multi-index notation](https://bliptext.com/articles/multi-index-notation) the Taylor series for several variables becomes\nwhich is to be understood as a still more abbreviated [multi-index](https://bliptext.com/articles/multi-index) version of the first equation of this paragraph, with a full analogy to the single variable case.\n\n## Example\n[[Image:Second Order Taylor.svg|200px|thumb|right|Second-order Taylor series approximation (in orange) of a function\nf (x,y) = ex ln(1 + y)\naround the origin.]]\nIn order to compute a second-order Taylor series expansion around point\n(a, b) = (0, 0)\nof the function\none first computes all the necessary partial derivatives:\nEvaluating these derivatives at the origin gives the Taylor coefficients\nSubstituting these values in to the general formula\nproduces\nSince\nln(1 + y)\nis analytic in\n, we have\n\n# Comparison with Fourier series\nThe trigonometric [Fourier series](https://bliptext.com/articles/fourier-series) enables one to express a [periodic function](https://bliptext.com/articles/periodic-function) (or a function defined on a closed interval\n[a,b]\n) as an infinite sum of [trigonometric functions](https://bliptext.com/articles/trigonometric-function) ([sines](https://bliptext.com/articles/sine) and co[sines](https://bliptext.com/articles/sine)). In this sense, the Fourier series is analogous to Taylor series, since the latter allows one to express a function as an infinite sum of [powers](https://bliptext.com/articles/power-function). Nevertheless, the two series differ from each other in several relevant issues:\nf (x)\nabout the point\nare all exactly equal to\nf\nat\na\n. In contrast, the Fourier series is computed by integrating over an entire interval, so there is generally no such point where all the finite truncations of the series are exact.\nf (x)\ncould be a [Weierstrass function](https://bliptext.com/articles/weierstrass-function).)","wordIndex":52,"createdAt":"2025-03-02T08:28:39.908Z","user":{"id":"YvtzH9WXGIETNdKmGpJ5vXsFRS2eiqs0","name":"disaster_code","image":"https://cdn.discordapp.com/embed/avatars/1.png"}}],"article":{"id":"50f4ee2f-8788-46b6-8b0b-9d4a3214be4b","title":"Taylor series","slug":"taylor-series","content":":::summary\n:::\n\n# \n[[File:sintay_SVG.svg|thumb|300px|As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows\nsin x\nand its Taylor approximations by polynomials of degree **1**, **3** , **5** , **7** , **9** , **1****1** , and **1****3** at\nx = 0\n.]]\nIn [mathematics](https://bliptext.com/articles/mathematics), the Swift or **Taylor expansion** of a [function](https://bliptext.com/articles/function-mathematics) is an [infinite sum](https://bliptext.com/articles/series-mathematics) of terms that are expressed in terms of the [function](https://bliptext.com/articles/function-mathematics)'s [derivatives](https://bliptext.com/articles/derivative) at a single point.\nFor most common functions, the function and the sum of Taylor Swift series are equal near this point.\nTaylor series are named after [Brook Taylor](https://bliptext.com/articles/brook-taylor), who introduced them in 1715.\nA Taylor series is also called a **Maclaurin series** when 0 is the point where the derivatives are considered, after [Colin Maclaurin](https://bliptext.com/articles/colin-maclaurin), who made extensive use of this special case of Taylor series in the 18th century.\nThe [partial sum](https://bliptext.com/articles/partial-sum) formed by the first\nn + 1\nterms of a Taylor series is a [polynomial](https://bliptext.com/articles/polynomial) of degree n that is called the nth **Taylor [polynomial](https://bliptext.com/articles/polynomial)** of the function.\nTaylor polynomials are approximations of a function, which become generally more accurate as n increases.\n[Taylor's theorem](https://bliptext.com/articles/taylor-s-theorem) gives quantitative estimates on the error introduced by the use of such approximations.\nIf the Taylor series of a function is [convergent](https://bliptext.com/articles/convergence-mathematics), its sum is the [limit](https://bliptext.com/articles/limit-of-a-sequence) of the [infinite sequence](https://bliptext.com/articles/infinite-sequence) of the Taylor polynomials.\nA function may differ from the sum of its Taylor series, even if its Taylor series is convergent.\nA function is [analytic](https://bliptext.com/articles/analytic-function) at a point x if it is equal to the sum of its Taylor series in some [open interval](https://bliptext.com/articles/open-interval) (or [open disk](https://bliptext.com/articles/disk-mathematics) in the [complex plane](https://bliptext.com/articles/complex-plane)) containing x. This implies that the function is [analytic](https://bliptext.com/articles/analytic-function) at every point of the interval (or disk).\n\n# Definition\nThe Taylor series of a [real](https://bliptext.com/articles/real-valued-function) or [complex-valued function](https://bliptext.com/articles/complex-valued-function)\nf (x)\n, that is [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable-function) at a [real](https://bliptext.com/articles/real-number) or [complex number](https://bliptext.com/articles/complex-number)\na\n, is the [power series](https://bliptext.com/articles/power-series)\nHere,\nn!\ndenotes the [factorial](https://bliptext.com/articles/factorial) of n. The function\nf(a)\ndenotes the nth [derivative](https://bliptext.com/articles/derivative) of f evaluated at the point a. The [derivative](https://bliptext.com/articles/derivative) of order zero of f is defined to be f itself and\n(x − a)0\nand\n0!\n[are both defined to be 1](https://bliptext.com/articles/empty-product).\nThis series can be written by using [sigma notation](https://bliptext.com/articles/sigma-notation), as in the right side formula.\nWith\n, the Maclaurin series takes the form:\n\n# Examples\nThe Taylor series of any [polynomial](https://bliptext.com/articles/polynomial) is the [polynomial](https://bliptext.com/articles/polynomial) itself.\nThe Maclaurin series of\n1⁄1 − x\nis the [geometric series](https://bliptext.com/articles/geometric-series)\nSo, by substituting x for\n1 − x\n, the Taylor series of\n1⁄x\nat\nis\nBy integrating the above Maclaurin series, we find the Maclaurin series of\nln(1 − x)\n, where\nln\ndenotes the [natural logarithm](https://bliptext.com/articles/natural-logarithm):\nThe corresponding Taylor series of\nln x\nat\nis\nand more generally, the corresponding Taylor series of\nln x\nat an arbitrary nonzero point a is:\nThe Maclaurin series of the [exponential function](https://bliptext.com/articles/exponential-function)\nex\nis\nThe above expansion holds because the derivative of\nex\nwith respect to x is also\nex\n, and\ne0\nequals 1.\nThis leaves the terms\n(x − 0)n\nin the numerator and\nn!\nin the denominator of each term in the infinite sum.\n\n# History\nThe [ancient Greek philosopher](https://bliptext.com/articles/ancient-greek-philosopher) [Zeno of Elea](https://bliptext.com/articles/zeno-of-elea) considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility; the result was [Zeno's paradox](https://bliptext.com/articles/zeno-s-paradox).\nLater, [Aristotle](https://bliptext.com/articles/aristotle) proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by [Archimedes](https://bliptext.com/articles/archimedes), as it had been prior to [Aristotle](https://bliptext.com/articles/aristotle) by the Presocratic Atomist [Democritus](https://bliptext.com/articles/democritus).\nIt was through Archimedes's [method of exhaustion](https://bliptext.com/articles/method-of-exhaustion) that an infinite number of progressive subdivisions could be performed to achieve a finite result.\n[Liu Hui](https://bliptext.com/articles/liu-hui) independently employed a similar method a few centuries later.\nIn the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by Indian mathematician [Madhava of Sangamagrama](https://bliptext.com/articles/madhava-of-sangamagrama).\nThough no record of his work survives, writings of his followers in the [Kerala school of astronomy and mathematics](https://bliptext.com/articles/kerala-school-of-astronomy-and-mathematics) suggest that he found the Taylor series for the [trigonometric functions](https://bliptext.com/articles/trigonometric-function) of [sine](https://bliptext.com/articles/sine), co[sine](https://bliptext.com/articles/sine), and [arctangent](https://bliptext.com/articles/arctangent) (see [Madhava series](https://bliptext.com/articles/madhava-series)).\nDuring the following two centuries his followers developed further series expansions and rational approximations.\nIn late 1670, [James Gregory](https://bliptext.com/articles/james-gregory-mathematician) was shown in a letter from [John Collins](https://bliptext.com/articles/john-collins-mathematician) several Maclaurin series (\\sin x, \\cos x, \\arcsin x, and x \\cot x) derived by [Isaac Newton](https://bliptext.com/articles/isaac-newton), and told that Newton had developed a general method for expanding functions in series.\nNewton had in fact used a cumbersome method involving long division of series and term-by-term integration, but Gregory did not know it and set out to discover a general method for himself.\nIn early 1671 Gregory discovered something like the general Maclaurin series and sent a letter to Collins including series for \\arctan x, \\tan x, \\sec x, (the integral of \\tan), (the [[integral of the secant function|integral of\nsec\n]], the inverse [Gudermannian function](https://bliptext.com/articles/gudermannian-function)), and (the [Gudermannian function](https://bliptext.com/articles/gudermannian-function)). However, thinking that he had merely redeveloped a method by Newton, Gregory never described how he obtained these series, and it can only be inferred that he understood the general method by examining scratch work he had scribbled on the back of another letter from 1671.\nIn 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work De Quadratura Curvarum.\nHowever, this work was never completed and the relevant sections were omitted from the portions published in 1704 under the title Tractatus de Quadratura Curvarum.\nIt was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by [Brook Taylor](https://bliptext.com/articles/brook-taylor), after whom the series are now named.\nThe Maclaurin series was named after [Colin Maclaurin](https://bliptext.com/articles/colin-maclaurin), a Scottish mathematician, who published the special case of the Taylor result in the mid-18th century.\n\n# Analytic functions\n[[Image:Exp neg inverse square.svg|upright=1.4|thumb|right|The function\ne(−1/x 2)\nis not analytic at\nx = 0\nIf\nf (x)\nis given by a convergent power series in an open disk centred at b in the complex plane (or an interval in the real line), it is said to be [analytic](https://bliptext.com/articles/analytic-function) in this region.\nThus for x in this region, f is given by a convergent power series\nDifferentiating by x the above formula n times, then setting\ngives:\nand so the power series expansion agrees with the Taylor series.\nThus a function is analytic in an open disk centered at b if and only if its Taylor series converges to the value of the function at each point of the disk.\nIf\nf (x)\nis equal to the sum of its Taylor series for all x in the complex plane, it is called [entire](https://bliptext.com/articles/entire-function).\nThe polynomials, [exponential function](https://bliptext.com/articles/exponential-function)\nex\n, and the [trigonometric functions](https://bliptext.com/articles/trigonometric-function) sine and cosine, are examples of entire functions.\nExamples of functions that are not entire include the [square root](https://bliptext.com/articles/square-root), the [logarithm](https://bliptext.com/articles/logarithm), the [trigonometric function](https://bliptext.com/articles/trigonometric-function) tangent, and its inverse, [arctan](https://bliptext.com/articles/arctan).\nFor these functions the Taylor series do not [converge](https://bliptext.com/articles/convergent-series) if x is far from b. That is, the Taylor series [diverges](https://bliptext.com/articles/divergent-series) at x if the distance between x and b is larger than the radius of [converge](https://bliptext.com/articles/convergent-series)nce.\nThe Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point.\nUses of the Taylor series for analytic functions include:\n\n# Approximation error and convergence\n[[Image:LogTay.svg|upright=1.4|thumb|right|The Taylor polynomials for\nln(1 + x)\nonly provide accurate approximations in the range\n−1 < x ≤ 1\n. For\nx > 1\n, Taylor polynomials of higher degree provide worse approximations.]]\n[[Image:Logarithm GIF.gif|upright=1.4|thumb|right|The Taylor approximations for\nln(1 + x)\n(black).\nFor\nx > 1\n, the approximations diverge.]]\nPictured is an accurate approximation of\nsin x\naround the point\n. The pink curve is a polynomial of degree seven:\nThe error in this approximation is no more than\n. For a full cycle centered at the origin (\n−π < x < π\n) the error is less than 0.08215. In particular, for\n−1 < x < 1\n, the error is less than 0.000003.\nIn contrast, also shown is a picture of the natural logarithm function\nln(1 + x)\nand some of its Taylor polynomials around\n. These approximations converge to the function only in the region\n−1 < x ≤ 1\nThe error incurred in approximating a function by its nth-degree Taylor polynomial is called the remainder or [residual](https://bliptext.com/articles/residual-numerical-analysis) and is denoted by the function\nRn(x)\n. Taylor's theorem can be used to obtain a bound on the [size of the remainder](https://bliptext.com/articles/taylor-s-theorem).\nIn general, Taylor series need not be [convergent](https://bliptext.com/articles/convergent-series) at all.\nIn fact, the set of functions with a convergent Taylor series is a [meager set](https://bliptext.com/articles/meager-set) in the [Fréchet space](https://bliptext.com/articles/fr-chet-space) of [smooth functions](https://bliptext.com/articles/smooth-functions).\nEven if the Taylor series of a function f does converge, its limit need not be equal to the value of the function\nf (x)\n. For example, the function\nis [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable) at\n, and has all derivatives zero there.\nConsequently, the Taylor series of\nf (x)\nabout\nis identically zero.\nHowever,\nf (x)\nis not the zero function, so does not equal its Taylor series around the origin.\nThus,\nf (x)\nis an example of a [non-analytic smooth function](https://bliptext.com/articles/non-analytic-smooth-function).\nIn [real analysis](https://bliptext.com/articles/real-analysis), this example shows that there are [infinitely differentiable functions](https://bliptext.com/articles/infinitely-differentiable-function)\nf (x)\nwhose Taylor series are not equal to\nf (x)\neven if they converge.\nBy contrast, the [holomorphic functions](https://bliptext.com/articles/holomorphic-function) studied in [complex analysis](https://bliptext.com/articles/complex-analysis) always possess a convergent Taylor series, and even the Taylor series of [meromorphic functions](https://bliptext.com/articles/meromorphic-function), which might have singularities, never converge to a value different from the function itself.\nThe complex function\ne−1/z 2\n, however, does not approach 0 when z approaches 0 along the imaginary axis, so it is not [continuous](https://bliptext.com/articles/continuous-function) in the complex plane and its Taylor series is undefined at 0.\nMore generally, every sequence of real or complex numbers can appear as [coefficients](https://bliptext.com/articles/coefficient) in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of [Borel's lemma](https://bliptext.com/articles/borel-s-lemma).\nAs a result, the [radius of convergence](https://bliptext.com/articles/radius-of-convergence) of a Taylor series can be zero.\nThere are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere.\nA function cannot be written as a Taylor series centred at a [singularity](https://bliptext.com/articles/singularity-mathematics); in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see [Laurent series](https://bliptext.com/articles/laurent-series).\nFor example,\nf (x) = e−1/x 2\ncan be written as a Laurent series.\n# [The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin. | upload.wikimedia.org/wikipedia/commons/b/b6/Taylorsine.svg]\n\n## Generalization\nThe generalization of the Taylor series does converge to the value of the function itself for any [bounded](https://bliptext.com/articles/bounded-function) [continuous function](https://bliptext.com/articles/continuous-function) on\n(0,∞)\n, and this can be done by using the calculus of [finite differences](https://bliptext.com/articles/finite-differences).\nSpecifically, the following theorem, due to [Einar Hille](https://bliptext.com/articles/einar-hille), that for any\nt > 0\n,\nHere\nΔn h\nis the nth finite difference operator with step size h. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the [Newton series](https://bliptext.com/articles/newton-series).\nWhen the function f is analytic at a, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series.\nIn general, for any infinite sequence\nai\n, the following power series identity holds:\nSo in particular,\nThe series on the right is the [expected value](https://bliptext.com/articles/expected-value) of\nf (a + X)\n, where X is a [Poisson-distributed](https://bliptext.com/articles/poisson-distribution) [random variable](https://bliptext.com/articles/random-variable) that takes the value\njh\nwith probability\ne−t/h·(t/h)⁄j!\n. Hence,\nThe [law of large numbers](https://bliptext.com/articles/law-of-large-numbers) implies that the identity holds.\n\n# List of Maclaurin series of some common functions\nSeveral important Maclaurin series expansions follow.\nAll these expansions are valid for complex arguments x.\n\n## Exponential function\nExp series.gif\nex\n(in blue), and the sum of the first\nn + 1\nterms of its Taylor series at 0 (in red).]]\nThe [exponential function](https://bliptext.com/articles/exponential-function) e^x (with base e) has Maclaurin series\nIt converges for all x.\nThe exponential [generating function](https://bliptext.com/articles/generating-function) of the [Bell numbers](https://bliptext.com/articles/bell-number) is the exponential function of the predecessor of the exponential function:\n\n## Natural logarithm\nThe [natural logarithm](https://bliptext.com/articles/natural-logarithm) (with base e) has Maclaurin series\nThe last series is known as [Mercator series](https://bliptext.com/articles/mercator-series), named after [Nicholas Mercator](https://bliptext.com/articles/nicholas-mercator) (since it was published in his 1668 treatise Logarithmotechnia).\nBoth of these series converge for |x| < 1.\n(In addition, the series for\nln(1 − x)\nconverges for\n, and the series for\nln(1 + x)\nconverges for\n.)\n\n## Geometric series\nThe [geometric series](https://bliptext.com/articles/geometric-series) and its derivatives have Maclaurin series\nAll are convergent for |x| < 1.\nThese are special cases of the [binomial series](https://bliptext.com/articles/) given in the next section.\n\n## Binomial series\nThe [binomial series](https://bliptext.com/articles/binomial-series) is the power series\nwhose coefficients are the generalized [binomial coefficients](https://bliptext.com/articles/binomial-coefficient)\n(If\nn = 0\n, this product is an [empty product](https://bliptext.com/articles/empty-product) and has value 1.) It converges for |x| < 1 for any real or complex number α.\nWhen\nα = −1\n, this is essentially the infinite geometric series mentioned in the previous section.\nThe special cases\nα = 1⁄2\nand\nα = −1⁄2\ngive the [square root](https://bliptext.com/articles/square-root) function and its [inverse](https://bliptext.com/articles/multiplicative-inverse):\nWhen only the [linear term](https://bliptext.com/articles/linear-approximation) is retained, this simplifies to the [binomial approximation](https://bliptext.com/articles/binomial-approximation).\n\n## Trigonometric functions\nThe usual [trigonometric functions](https://bliptext.com/articles/trigonometric-function) and their inverses have the following Maclaurin series:\nAll angles are expressed in [radians](https://bliptext.com/articles/radian).\nThe numbers\nBk\nappearing in the expansions of\ntan x\nare the [Bernoulli numbers](https://bliptext.com/articles/bernoulli-numbers).\nThe\nEk\nin the expansion of\nsec x\nare [Euler numbers](https://bliptext.com/articles/euler-number).\n\n## Hyperbolic functions\nThe [hyperbolic functions](https://bliptext.com/articles/hyperbolic-function) have Maclaurin series closely related to the series for the corresponding trigonometric functions:\nThe numbers\nBk\nappearing in the series for\ntanh x\nare the [Bernoulli numbers](https://bliptext.com/articles/bernoulli-numbers).\n\n## Polylogarithmic functions\nThe [polylogarithms](https://bliptext.com/articles/polylogarithm) have these defining identities:\nThe [Legendre chi functions](https://bliptext.com/articles/legendre-chi-function) are defined as follows:\nAnd the formulas presented below are called [inverse tangent integrals](https://bliptext.com/articles/inverse-tangent-integral):\nIn [statistical thermodynamics](https://bliptext.com/articles/statistical-mechanics) these formulas are of great importance.\n\n## Elliptic functions\nThe complete [elliptic integrals](https://bliptext.com/articles/elliptic-integral) of first kind K and of second kind E can be defined as follows:\nThe [Jacobi theta functions](https://bliptext.com/articles/theta-function) describe the world of the elliptic modular functions and they have these Taylor series:\nThe regular [partition number sequence](https://bliptext.com/articles/partition-function-number-theory) P(n) has this generating function:\nThe strict partition number sequence Q(n) has that generating function:\n\n# Calculation of Taylor series\nSeveral methods exist for the calculation of Taylor series of a large number of functions.\nOne can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern.\nAlternatively, one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series.\nIn some cases, one can also derive the Taylor series by repeatedly applying [integration by parts](https://bliptext.com/articles/integration-by-parts).\nParticularly convenient is the use of [computer algebra systems](https://bliptext.com/articles/computer-algebra-system) to calculate Taylor series.\n\n## First example\nIn order to compute the 7th degree Maclaurin polynomial for the function\none may first rewrite the function as\nthe composition of two functions and The Taylor series for the natural logarithm is (using [big O notation](https://bliptext.com/articles/big-o-notation))\nand for the cosine function\nThe first several terms from the second series can be substituted into each term of the first series.\nBecause the first term in the second series has degree 2, three terms of the first series suffice to give a 7th-degree polynomial:\nSince the cosine is an [even function](https://bliptext.com/articles/even-function), the coefficients for all the odd powers are zero.\n\n## Second example\nSuppose we want the Taylor series at 0 of the function\nThe Taylor series for the exponential function is\nand the series for cosine is\nAssume the series for their quotient is\nMultiplying both sides by the denominator \\cos x and then expanding it as a series yields\nComparing the coefficients of g(x)\\cos x with the coefficients of e^x,\nThe coefficients c_i of the series for g(x) can thus be computed one at a time, amounting to long division of the series for e^x and \\cos x:\n\n## Third example\nHere we employ a method called \"indirect expansion\" to expand the given function.\nThis method uses the known Taylor expansion of the exponential function.\nIn order to expand\n(1 + x)ex\nas a Taylor series in x, we use the known Taylor series of function\nex\nThus,\n\n# Taylor series as definitions\nClassically, [algebraic functions](https://bliptext.com/articles/algebraic-function) are defined by an algebraic equation, and [transcendental functions](https://bliptext.com/articles/transcendental-function) (including those discussed above) are defined by some property that holds for them, such as a [differential equation](https://bliptext.com/articles/differential-equation).\nFor example, the [exponential function](https://bliptext.com/articles/exponential-function) is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin.\nHowever, one may equally well define an [analytic function](https://bliptext.com/articles/analytic-function) by its Taylor series.\nTaylor series are used to define functions and \"[operators](https://bliptext.com/articles/operator-mathematics)\" in diverse areas of mathematics.\nIn particular, this is true in areas where the classical definitions of functions break down.\nFor example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the [matrix exponential](https://bliptext.com/articles/matrix-exponential) or [matrix logarithm](https://bliptext.com/articles/matrix-logarithm).\nIn other areas, such as formal analysis, it is more convenient to work directly with the [power series](https://bliptext.com/articles/power-series) themselves.\nThus one may define a solution of a differential equation as a power series which, one hopes to prove, is the Taylor series of the desired solution.\n\n# Taylor series in several variables\nThe Taylor series may also be generalized to functions of more than one variable with\nFor example, for a function f(x,y) that depends on two variables, x and y, the Taylor series to second order about the point\n(a, b)\nis\nwhere the subscripts denote the respective [partial derivatives](https://bliptext.com/articles/partial-derivative).\n\n## Second-order Taylor series in several variables\nA second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as\nwhere\nD f (a)\nis the [gradient](https://bliptext.com/articles/gradient) of f evaluated at\nand\nD2 f (a)\nis the [Hessian matrix](https://bliptext.com/articles/hessian-matrix).\nApplying the [multi-index notation](https://bliptext.com/articles/multi-index-notation) the Taylor series for several variables becomes\nwhich is to be understood as a still more abbreviated [multi-index](https://bliptext.com/articles/multi-index) version of the first equation of this paragraph, with a full analogy to the single variable case.\n\n## Example\n[[Image:Second Order Taylor.svg|200px|thumb|right|Second-order Taylor series approximation (in orange) of a function\nf (x,y) = ex ln(1 + y)\naround the origin.]]\nIn order to compute a second-order Taylor series expansion around point\n(a, b) = (0, 0)\nof the function\none first computes all the necessary partial derivatives:\nEvaluating these derivatives at the origin gives the Taylor coefficients\nSubstituting these values in to the general formula\nproduces\nSince\nln(1 + y)\nis analytic in\n, we have\n\n# Comparison with Fourier series\nThe trigonometric [Fourier series](https://bliptext.com/articles/fourier-series) enables one to express a [periodic function](https://bliptext.com/articles/periodic-function) (or a function defined on a closed interval\n[a,b]\n) as an infinite sum of [trigonometric functions](https://bliptext.com/articles/trigonometric-function) ([sines](https://bliptext.com/articles/sine) and co[sines](https://bliptext.com/articles/sine)). In this sense, the Fourier series is analogous to Taylor series, since the latter allows one to express a function as an infinite sum of [powers](https://bliptext.com/articles/power-function). Nevertheless, the two series differ from each other in several relevant issues:\nf (x)\nabout the point\nare all exactly equal to\nf\nat\na\n. In contrast, the Fourier series is computed by integrating over an entire interval, so there is generally no such point where all the finite truncations of the series are exact.\nf (x)\ncould be a [Weierstrass function](https://bliptext.com/articles/weierstrass-function).)","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T16:47:08.206Z","updated_at":"2025-04-24T00:59:38.591Z","search_vector":"'seri':2 'taylor':1","search_content":" \n \n\n# \n[[File:sintay_SVG.svg|thumb|300px|As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows\nsin x\nand its Taylor approximations by polynomials of degree **1**, **3** , **5** , **7** , **9** , **1****1** , and **1****3** at\nx = 0\n.]]\nIn [mathematics](https://bliptext.com/articles/mathematics), the **Taylor series** or **Taylor expansion** of a [function](https://bliptext.com/articles/function-mathematics) is an [infinite sum](https://bliptext.com/articles/series-mathematics) of terms that are expressed in terms of the [function](https://bliptext.com/articles/function-mathematics)'s [derivatives](https://bliptext.com/articles/derivative) at a single point.\nFor most common functions, the function and the sum of its Taylor series are equal near this point.\nTaylor series are named after [Brook Taylor](https://bliptext.com/articles/brook-taylor), who introduced them in 1715.\nA Taylor series is also called a **Maclaurin series** when 0 is the point where the derivatives are considered, after [Colin Maclaurin](https://bliptext.com/articles/colin-maclaurin), who made extensive use of this special case of Taylor series in the 18th century.\nThe [partial sum](https://bliptext.com/articles/partial-sum) formed by the first\nn + 1\nterms of a Taylor series is a [polynomial](https://bliptext.com/articles/polynomial) of degree n that is called the nth **Taylor [polynomial](https://bliptext.com/articles/polynomial)** of the function.\nTaylor polynomials are approximations of a function, which become generally more accurate as n increases.\n[Taylor's theorem](https://bliptext.com/articles/taylor-s-theorem) gives quantitative estimates on the error introduced by the use of such approximations.\nIf the Taylor series of a function is [convergent](https://bliptext.com/articles/convergence-mathematics), its sum is the [limit](https://bliptext.com/articles/limit-of-a-sequence) of the [infinite sequence](https://bliptext.com/articles/infinite-sequence) of the Taylor polynomials.\nA function may differ from the sum of its Taylor series, even if its Taylor series is convergent.\nA function is [analytic](https://bliptext.com/articles/analytic-function) at a point x if it is equal to the sum of its Taylor series in some [open interval](https://bliptext.com/articles/open-interval) (or [open disk](https://bliptext.com/articles/disk-mathematics) in the [complex plane](https://bliptext.com/articles/complex-plane)) containing x. This implies that the function is [analytic](https://bliptext.com/articles/analytic-function) at every point of the interval (or disk).\n\n# Definition\nThe Taylor series of a [real](https://bliptext.com/articles/real-valued-function) or [complex-valued function](https://bliptext.com/articles/complex-valued-function)\nf (x)\n, that is [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable-function) at a [real](https://bliptext.com/articles/real-number) or [complex number](https://bliptext.com/articles/complex-number)\na\n, is the [power series](https://bliptext.com/articles/power-series)\nHere,\nn!\ndenotes the [factorial](https://bliptext.com/articles/factorial) of n. The function\nf(a)\ndenotes the nth [derivative](https://bliptext.com/articles/derivative) of f evaluated at the point a. The [derivative](https://bliptext.com/articles/derivative) of order zero of f is defined to be f itself and\n(x − a)0\nand\n0!\n[are both defined to be 1](https://bliptext.com/articles/empty-product).\nThis series can be written by using [sigma notation](https://bliptext.com/articles/sigma-notation), as in the right side formula.\nWith\n, the Maclaurin series takes the form:\n\n# Examples\nThe Taylor series of any [polynomial](https://bliptext.com/articles/polynomial) is the [polynomial](https://bliptext.com/articles/polynomial) itself.\nThe Maclaurin series of\n1⁄1 − x\nis the [geometric series](https://bliptext.com/articles/geometric-series)\nSo, by substituting x for\n1 − x\n, the Taylor series of\n1⁄x\nat\nis\nBy integrating the above Maclaurin series, we find the Maclaurin series of\nln(1 − x)\n, where\nln\ndenotes the [natural logarithm](https://bliptext.com/articles/natural-logarithm):\nThe corresponding Taylor series of\nln x\nat\nis\nand more generally, the corresponding Taylor series of\nln x\nat an arbitrary nonzero point a is:\nThe Maclaurin series of the [exponential function](https://bliptext.com/articles/exponential-function)\nex\nis\nThe above expansion holds because the derivative of\nex\nwith respect to x is also\nex\n, and\ne0\nequals 1.\nThis leaves the terms\n(x − 0)n\nin the numerator and\nn!\nin the denominator of each term in the infinite sum.\n\n# History\nThe [ancient Greek philosopher](https://bliptext.com/articles/ancient-greek-philosopher) [Zeno of Elea](https://bliptext.com/articles/zeno-of-elea) considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility; the result was [Zeno's paradox](https://bliptext.com/articles/zeno-s-paradox).\nLater, [Aristotle](https://bliptext.com/articles/aristotle) proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by [Archimedes](https://bliptext.com/articles/archimedes), as it had been prior to [Aristotle](https://bliptext.com/articles/aristotle) by the Presocratic Atomist [Democritus](https://bliptext.com/articles/democritus).\nIt was through Archimedes's [method of exhaustion](https://bliptext.com/articles/method-of-exhaustion) that an infinite number of progressive subdivisions could be performed to achieve a finite result.\n[Liu Hui](https://bliptext.com/articles/liu-hui) independently employed a similar method a few centuries later.\nIn the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by Indian mathematician [Madhava of Sangamagrama](https://bliptext.com/articles/madhava-of-sangamagrama).\nThough no record of his work survives, writings of his followers in the [Kerala school of astronomy and mathematics](https://bliptext.com/articles/kerala-school-of-astronomy-and-mathematics) suggest that he found the Taylor series for the [trigonometric functions](https://bliptext.com/articles/trigonometric-function) of [sine](https://bliptext.com/articles/sine), co[sine](https://bliptext.com/articles/sine), and [arctangent](https://bliptext.com/articles/arctangent) (see [Madhava series](https://bliptext.com/articles/madhava-series)).\nDuring the following two centuries his followers developed further series expansions and rational approximations.\nIn late 1670, [James Gregory](https://bliptext.com/articles/james-gregory-mathematician) was shown in a letter from [John Collins](https://bliptext.com/articles/john-collins-mathematician) several Maclaurin series (\\sin x, \\cos x, \\arcsin x, and x \\cot x) derived by [Isaac Newton](https://bliptext.com/articles/isaac-newton), and told that Newton had developed a general method for expanding functions in series.\nNewton had in fact used a cumbersome method involving long division of series and term-by-term integration, but Gregory did not know it and set out to discover a general method for himself.\nIn early 1671 Gregory discovered something like the general Maclaurin series and sent a letter to Collins including series for \\arctan x, \\tan x, \\sec x, (the integral of \\tan), (the [[integral of the secant function|integral of\nsec\n]], the inverse [Gudermannian function](https://bliptext.com/articles/gudermannian-function)), and (the [Gudermannian function](https://bliptext.com/articles/gudermannian-function)). However, thinking that he had merely redeveloped a method by Newton, Gregory never described how he obtained these series, and it can only be inferred that he understood the general method by examining scratch work he had scribbled on the back of another letter from 1671.\nIn 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work De Quadratura Curvarum.\nHowever, this work was never completed and the relevant sections were omitted from the portions published in 1704 under the title Tractatus de Quadratura Curvarum.\nIt was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by [Brook Taylor](https://bliptext.com/articles/brook-taylor), after whom the series are now named.\nThe Maclaurin series was named after [Colin Maclaurin](https://bliptext.com/articles/colin-maclaurin), a Scottish mathematician, who published the special case of the Taylor result in the mid-18th century.\n\n# Analytic functions\n[[Image:Exp neg inverse square.svg|upright=1.4|thumb|right|The function\n e(−1/x 2)\nis not analytic at\nx = 0\nIf\nf (x)\nis given by a convergent power series in an open disk centred at b in the complex plane (or an interval in the real line), it is said to be [analytic](https://bliptext.com/articles/analytic-function) in this region.\nThus for x in this region, f is given by a convergent power series\nDifferentiating by x the above formula n times, then setting\ngives:\nand so the power series expansion agrees with the Taylor series.\nThus a function is analytic in an open disk centered at b if and only if its Taylor series converges to the value of the function at each point of the disk.\nIf\nf (x)\nis equal to the sum of its Taylor series for all x in the complex plane, it is called [entire](https://bliptext.com/articles/entire-function).\nThe polynomials, [exponential function](https://bliptext.com/articles/exponential-function)\nex\n, and the [trigonometric functions](https://bliptext.com/articles/trigonometric-function) sine and cosine, are examples of entire functions.\nExamples of functions that are not entire include the [square root](https://bliptext.com/articles/square-root), the [logarithm](https://bliptext.com/articles/logarithm), the [trigonometric function](https://bliptext.com/articles/trigonometric-function) tangent, and its inverse, [arctan](https://bliptext.com/articles/arctan).\nFor these functions the Taylor series do not [converge](https://bliptext.com/articles/convergent-series) if x is far from b. That is, the Taylor series [diverges](https://bliptext.com/articles/divergent-series) at x if the distance between x and b is larger than the radius of [converge](https://bliptext.com/articles/convergent-series)nce.\nThe Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point.\nUses of the Taylor series for analytic functions include:\n\n# Approximation error and convergence\n[[Image:LogTay.svg|upright=1.4|thumb|right|The Taylor polynomials for\nln(1 + x)\nonly provide accurate approximations in the range\n−1 1\n, Taylor polynomials of higher degree provide worse approximations.]]\n[[Image:Logarithm GIF.gif|upright=1.4|thumb|right|The Taylor approximations for\nln(1 + x)\n(black).\nFor\nx > 1\n, the approximations diverge.]]\nPictured is an accurate approximation of\nsin x\naround the point\n. The pink curve is a polynomial of degree seven:\nThe error in this approximation is no more than\n. For a full cycle centered at the origin (\n−π 0\n,\nHere\nΔn h\nis the nth finite difference operator with step size h. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the [Newton series](https://bliptext.com/articles/newton-series).\nWhen the function f is analytic at a, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series.\nIn general, for any infinite sequence\nai\n, the following power series identity holds:\nSo in particular,\nThe series on the right is the [expected value](https://bliptext.com/articles/expected-value) of\nf (a + X)\n, where X is a [Poisson-distributed](https://bliptext.com/articles/poisson-distribution) [random variable](https://bliptext.com/articles/random-variable) that takes the value\njh\nwith probability\ne−t/h·(t/h)⁄j!\n. Hence,\nThe [law of large numbers](https://bliptext.com/articles/law-of-large-numbers) implies that the identity holds.\n\n# List of Maclaurin series of some common functions\nSeveral important Maclaurin series expansions follow.\nAll these expansions are valid for complex arguments x.\n\n## Exponential function\nExp series.gif\nex\n(in blue), and the sum of the first\nn + 1\nterms of its Taylor series at 0 (in red).]]\nThe [exponential function](https://bliptext.com/articles/exponential-function) e^x (with base e) has Maclaurin series\nIt converges for all x.\nThe exponential [generating function](https://bliptext.com/articles/generating-function) of the [Bell numbers](https://bliptext.com/articles/bell-number) is the exponential function of the predecessor of the exponential function:\n\n## Natural logarithm\nThe [natural logarithm](https://bliptext.com/articles/natural-logarithm) (with base e) has Maclaurin series\nThe last series is known as [Mercator series](https://bliptext.com/articles/mercator-series), named after [Nicholas Mercator](https://bliptext.com/articles/nicholas-mercator) (since it was published in his 1668 treatise Logarithmotechnia).\nBoth of these series converge for |x| < 1.\n(In addition, the series for\nln(1 − x)\nconverges for\n, and the series for\nln(1 + x)\nconverges for\n.)\n\n## Geometric series\nThe [geometric series](https://bliptext.com/articles/geometric-series) and its derivatives have Maclaurin series\nAll are convergent for |x| < 1.\nThese are special cases of the [binomial series](https://bliptext.com/articles/) given in the next section.\n\n## Binomial series\nThe [binomial series](https://bliptext.com/articles/binomial-series) is the power series\nwhose coefficients are the generalized [binomial coefficients](https://bliptext.com/articles/binomial-coefficient)\n(If\nn = 0\n, this product is an [empty product](https://bliptext.com/articles/empty-product) and has value 1.) It converges for |x| < 1 for any real or complex number α.\nWhen\nα = −1\n, this is essentially the infinite geometric series mentioned in the previous section.\nThe special cases\nα = 1⁄2\nand\nα = −1⁄2\ngive the [square root](https://bliptext.com/articles/square-root) function and its [inverse](https://bliptext.com/articles/multiplicative-inverse):\nWhen only the [linear term](https://bliptext.com/articles/linear-approximation) is retained, this simplifies to the [binomial approximation](https://bliptext.com/articles/binomial-approximation).\n\n## Trigonometric functions\nThe usual [trigonometric functions](https://bliptext.com/articles/trigonometric-function) and their inverses have the following Maclaurin series:\nAll angles are expressed in [radians](https://bliptext.com/articles/radian).\nThe numbers\nBk\nappearing in the expansions of\ntan x\nare the [Bernoulli numbers](https://bliptext.com/articles/bernoulli-numbers).\nThe\nEk\nin the expansion of\nsec x\nare [Euler numbers](https://bliptext.com/articles/euler-number).\n\n## Hyperbolic functions\nThe [hyperbolic functions](https://bliptext.com/articles/hyperbolic-function) have Maclaurin series closely related to the series for the corresponding trigonometric functions:\nThe numbers\nBk\nappearing in the series for\ntanh x\nare the [Bernoulli numbers](https://bliptext.com/articles/bernoulli-numbers).\n\n## Polylogarithmic functions\nThe [polylogarithms](https://bliptext.com/articles/polylogarithm) have these defining identities:\nThe [Legendre chi functions](https://bliptext.com/articles/legendre-chi-function) are defined as follows:\nAnd the formulas presented below are called [inverse tangent integrals](https://bliptext.com/articles/inverse-tangent-integral):\nIn [statistical thermodynamics](https://bliptext.com/articles/statistical-mechanics) these formulas are of great importance.\n\n## Elliptic functions\nThe complete [elliptic integrals](https://bliptext.com/articles/elliptic-integral) of first kind K and of second kind E can be defined as follows:\nThe [Jacobi theta functions](https://bliptext.com/articles/theta-function) describe the world of the elliptic modular functions and they have these Taylor series:\nThe regular [partition number sequence](https://bliptext.com/articles/partition-function-number-theory) P(n) has this generating function:\nThe strict partition number sequence Q(n) has that generating function:\n\n# Calculation of Taylor series\nSeveral methods exist for the calculation of Taylor series of a large number of functions.\nOne can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern.\nAlternatively, one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series.\nIn some cases, one can also derive the Taylor series by repeatedly applying [integration by parts](https://bliptext.com/articles/integration-by-parts).\nParticularly convenient is the use of [computer algebra systems](https://bliptext.com/articles/computer-algebra-system) to calculate Taylor series.\n\n## First example\nIn order to compute the 7th degree Maclaurin polynomial for the function\none may first rewrite the function as\nthe composition of two functions and The Taylor series for the natural logarithm is (using [big O notation](https://bliptext.com/articles/big-o-notation))\nand for the cosine function\nThe first several terms from the second series can be substituted into each term of the first series.\nBecause the first term in the second series has degree 2, three terms of the first series suffice to give a 7th-degree polynomial:\nSince the cosine is an [even function](https://bliptext.com/articles/even-function), the coefficients for all the odd powers are zero.\n\n## Second example\nSuppose we want the Taylor series at 0 of the function\nThe Taylor series for the exponential function is\nand the series for cosine is\nAssume the series for their quotient is\nMultiplying both sides by the denominator \\cos x and then expanding it as a series yields\nComparing the coefficients of g(x)\\cos x with the coefficients of e^x,\nThe coefficients c_i of the series for g(x) can thus be computed one at a time, amounting to long division of the series for e^x and \\cos x:\n\n## Third example\nHere we employ a method called \"indirect expansion\" to expand the given function.\nThis method uses the known Taylor expansion of the exponential function.\nIn order to expand\n(1 + x)ex\nas a Taylor series in x, we use the known Taylor series of function\nex\nThus,\n\n# Taylor series as definitions\nClassically, [algebraic functions](https://bliptext.com/articles/algebraic-function) are defined by an algebraic equation, and [transcendental functions](https://bliptext.com/articles/transcendental-function) (including those discussed above) are defined by some property that holds for them, such as a [differential equation](https://bliptext.com/articles/differential-equation).\nFor example, the [exponential function](https://bliptext.com/articles/exponential-function) is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin.\nHowever, one may equally well define an [analytic function](https://bliptext.com/articles/analytic-function) by its Taylor series.\nTaylor series are used to define functions and \"[operators](https://bliptext.com/articles/operator-mathematics)\" in diverse areas of mathematics.\nIn particular, this is true in areas where the classical definitions of functions break down.\nFor example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the [matrix exponential](https://bliptext.com/articles/matrix-exponential) or [matrix logarithm](https://bliptext.com/articles/matrix-logarithm).\nIn other areas, such as formal analysis, it is more convenient to work directly with the [power series](https://bliptext.com/articles/power-series) themselves.\nThus one may define a solution of a differential equation as a power series which, one hopes to prove, is the Taylor series of the desired solution.\n\n# Taylor series in several variables\nThe Taylor series may also be generalized to functions of more than one variable with\nFor example, for a function f(x,y) that depends on two variables, x and y, the Taylor series to second order about the point\n(a, b)\nis\nwhere the subscripts denote the respective [partial derivatives](https://bliptext.com/articles/partial-derivative).\n\n## Second-order Taylor series in several variables\nA second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as\nwhere\nD f (a)\nis the [gradient](https://bliptext.com/articles/gradient) of f evaluated at\nand\nD2 f (a)\nis the [Hessian matrix](https://bliptext.com/articles/hessian-matrix).\nApplying the [multi-index notation](https://bliptext.com/articles/multi-index-notation) the Taylor series for several variables becomes\nwhich is to be understood as a still more abbreviated [multi-index](https://bliptext.com/articles/multi-index) version of the first equation of this paragraph, with a full analogy to the single variable case.\n\n## Example\n[[Image:Second Order Taylor.svg|200px|thumb|right|Second-order Taylor series approximation (in orange) of a function\nf (x,y) = ex ln(1 + y)\naround the origin.]]\nIn order to compute a second-order Taylor series expansion around point\n(a, b) = (0, 0)\nof the function\none first computes all the necessary partial derivatives:\nEvaluating these derivatives at the origin gives the Taylor coefficients\nSubstituting these values in to the general formula\nproduces\nSince\nln(1 + y)\nis analytic in\n, we have\n\n# Comparison with Fourier series\nThe trigonometric [Fourier series](https://bliptext.com/articles/fourier-series) enables one to express a [periodic function](https://bliptext.com/articles/periodic-function) (or a function defined on a closed interval\n[a,b]\n) as an infinite sum of [trigonometric functions](https://bliptext.com/articles/trigonometric-function) ([sines](https://bliptext.com/articles/sine) and co[sines](https://bliptext.com/articles/sine)). In this sense, the Fourier series is analogous to Taylor series, since the latter allows one to express a function as an infinite sum of [powers](https://bliptext.com/articles/power-function). Nevertheless, the two series differ from each other in several relevant issues:\nf (x)\nabout the point\nare all exactly equal to\nf\nat\na\n. In contrast, the Fourier series is computed by integrating over an entire interval, so there is generally no such point where all the finite truncations of the series are exact.\nf (x)\ncould be a [Weierstrass function](https://bliptext.com/articles/weierstrass-function).)","revisionCount":3},"hasMore":false,"nextCursor":null}