{"history":[],"article":{"id":"0f52981e-0f6e-491e-a2a0-c22594ece6e9","title":"Taylor's theorem","slug":"taylor-s-theorem","content":":::summary\n:::\n\n# \nIn [calculus](https://bliptext.com/articles/calculus), **Taylor's theorem** gives an approximation of a k-times [differentiable function](https://bliptext.com/articles/differentiable-function) around a given point by a [polynomial](https://bliptext.com/articles/polynomial) of degree k, called the k-th-order **Taylor [polynomial](https://bliptext.com/articles/polynomial)**.\nFor a [smooth function](https://bliptext.com/articles/smooth-function), the Taylor polynomial is the truncation at the order k of the [Taylor series](https://bliptext.com/articles/taylor-series) of the function.\nThe first-order Taylor polynomial is the [linear approximation](https://bliptext.com/articles/linear-approximation) of the function, and the second-order Taylor polynomial is often referred to as the **quadratic approximation**.\nThere are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.\nTaylor's theorem is named after the mathematician [Brook Taylor](https://bliptext.com/articles/brook-taylor), who stated a version of it in 1715, although an earlier version of the result was already mentioned in [1671](https://bliptext.com/articles/1671-in-science) by [James Gregory](https://bliptext.com/articles/james-gregory-astronomer-and-mathematician).\nTaylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in [mathematical analysis](https://bliptext.com/articles/mathematical-analysis).\nIt gives simple arithmetic formulas to accurately compute values of many [transcendental functions](https://bliptext.com/articles/transcendental-function) such as the [exponential function](https://bliptext.com/articles/exponential-function) and [trigonometric functions](https://bliptext.com/articles/trigonometric-function).\nIt is the starting point of the study of [analytic functions](https://bliptext.com/articles/analytic-function), and is fundamental in various areas of mathematics, as well as in [numerical analysis](https://bliptext.com/articles/numerical-analysis) and [mathematical physics](https://bliptext.com/articles/mathematical-physics).\nTaylor's theorem also generalizes to [multivariate](https://bliptext.com/articles/multivariate-function) and [vector valued](https://bliptext.com/articles/vector-valued-function) functions.\nIt provided the mathematical basis for some landmark early computing machines: [Charles Babbage](https://bliptext.com/articles/charles-babbage)'s [Difference Engine](https://bliptext.com/articles/difference-engine) calculated sines, cosines, logarithms, and other transcendental functions by numerically integrating the first 7 terms of their Taylor series.\n# [300px | upload.wikimedia.org/wikipedia/commons/6/64/Taylorspolynomialexbig.svg]\n\n# Motivation\nIf a real-valued [function](https://bliptext.com/articles/function-mathematics) f(x) is [differentiable](https://bliptext.com/articles/derivative) at the point x=a, then it has a [linear approximation](https://bliptext.com/articles/linear-approximation) near this point.\nThis means that there exists a function h1(x) such that\nHere\nis the linear approximation of f(x) for x near the point a, whose graph y=P_1(x) is the [tangent line](https://bliptext.com/articles/tangent-line) to the graph y=f(x) at x = a. The error in the approximation is:\nAs x tends to a, this error goes to zero much faster than, making a useful approximation.\nFor a better approximation to f(x), we can fit a [quadratic polynomial](https://bliptext.com/articles/quadratic-polynomial) instead of a linear function:\nInstead of just matching one derivative of f(x) at x=a, this polynomial has the same first and second derivatives, as is evident upon differentiation.\nTaylor's theorem ensures that the quadratic approximation is, in a sufficiently small neighborhood of x=a, more accurate than the linear approximation.\nSpecifically,\nHere the error in the approximation is\nwhich, given the limiting behavior of h_2, goes to zero faster than (x - a)^2 as x tends to a.\nSimilarly, we might get still better approximations to f if we use [polynomials](https://bliptext.com/articles/polynomial) of higher degree, since then we can match even more derivatives with f at the selected base point.\nIn general, the error in approximating a function by a polynomial of degree k will go to zero much faster than (x-a)^k as x tends to a.\nHowever, there are functions, even infinitely differentiable ones, for which increasing the degree of the approximating polynomial does not increase the accuracy of approximation: we say such a function fails to be [analytic](https://bliptext.com/articles/analytic-function) at x = a: it is not (locally) determined by its derivatives at this point.\nTaylor's theorem is of asymptotic nature: it only tells us that the error R_k in an [approximation](https://bliptext.com/articles/approximation) by a k-th order Taylor [polynomial](https://bliptext.com/articles/polynomial) Pk tends to zero faster than any nonzero k-th degree [polynomial](https://bliptext.com/articles/polynomial) as x \\to a. It does not tell us how large the error is in any concrete [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) of the center of expansion, but for this purpose there are explicit formulas for the remainder term (given below) which are valid under some additional regularity assumptions on f.\nThese enhanced versions of Taylor's theorem typically lead to [uniform estimates](https://bliptext.com/articles/uniform-convergence) for the approximation error in a small neighborhood of the center of expansion, but the estimates do not necessarily hold for neighborhoods which are too large, even if the function f is [analytic](https://bliptext.com/articles/analytic-function).\nIn that situation one may have to select several Taylor polynomials with different centers of expansion to have reliable Taylor-approximations of the original function (see animation on the right.)\nThere are several ways we might use the remainder term:\n# [Taylor's theorem | upload.wikimedia.org/wikipedia/commons/6/6e/E%5Ex///with///linear///approximation.png]\n# [Taylor's theorem | upload.wikimedia.org/wikipedia/commons/6/6b/E%5Ex///with///quadratic///approximation///corrected.png]\n\n# Taylor's theorem in one real variable\n\n## Statement of the theorem\nThe precise statement of the most basic version of Taylor's theorem is as follows:\nThe polynomial appearing in Taylor's theorem is the **-th order Taylor polynomial**\nof the function f at the point a.\nThe Taylor polynomial is the unique \"asymptotic best fit\" polynomial in the sense that if there exists a function hk : R → R and a k-th order polynomial p such that\nthen p = Pk.\nTaylor's theorem describes the asymptotic behavior of the **remainder term**\nwhich is the [approximation error](https://bliptext.com/articles/approximation-error) when approximating f with its Taylor polynomial.\nUsing the [little-o notation](https://bliptext.com/articles/little-o-notation), the statement in Taylor's theorem reads as\n\n## Explicit formulas for the remainder\nUnder stronger regularity assumptions on f there are several precise formulas for the remainder term Rk of the Taylor polynomial, the most common ones being the following.\nThese refinements of Taylor's theorem are usually proved using the [mean value theorem](https://bliptext.com/articles/mean-value-theorem), whence the name.\nAdditionally, notice that this is precisely the [mean value theorem](https://bliptext.com/articles/mean-value-theorem) when k=0.\nAlso other similar expressions can be found.\nFor example, if G(t) is continuous on the closed interval and differentiable with a non-vanishing derivative on the open interval between a and x, then\nfor some number \\xi between a and x. This version covers the Lagrange and Cauchy forms of the remainder as special cases, and is proved below using [Cauchy's mean value theorem](https://bliptext.com/articles/mean-value-theorem).\nThe Lagrange form is obtained by taking and the Cauchy form is obtained by taking G(t)=t-a.\nThe statement for the integral form of the remainder is more advanced than the previous ones, and requires understanding of [Lebesgue integration theory](https://bliptext.com/articles/lebesgue-integral) for the full generality.\nHowever, it holds also in the sense of [Riemann integral](https://bliptext.com/articles/riemann-integral) provided the (k + 1)th derivative of f is continuous on the closed interval [a,x].\nDue to the [absolute continuity](https://bliptext.com/articles/absolutely-continuous) of f on the [closed interval](https://bliptext.com/articles/closed-interval) between a and x, its derivative f exists as an L-function, and the result can be [proven](https://bliptext.com/articles/) by a formal calculation using the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) and [integration by parts](https://bliptext.com/articles/integration-by-parts).\n\n## Estimates for the remainder\nIt is often useful in practice to be able to estimate the remainder term appearing in the Taylor approximation, rather than having an exact formula for it.\nSuppose that f is (k + 1)-times continuously differentiable in an interval I containing a.\nSuppose that there are real constants q and Q such that\nthroughout I.\nThen the remainder term satisfies the inequality\nif x > a, and a similar estimate if x < a. This is a simple consequence of the Lagrange form of the remainder.\nIn particular, if\non an interval I = (a − r,a + r) with some r > 0, then\nfor all x∈(a − r,a + r).\nThe second inequality is called a [uniform estimate](https://bliptext.com/articles/uniform-convergence), because it holds uniformly for all x on the interval (a − r,a + r).\n\n## Example\nSuppose that we wish to find the approximate value of the function f(x)=e^x on the interval [-1,1] while ensuring that the error in the approximation is no more than 10−5.\nIn this example we pretend that we only know the following properties of the exponential function:\nFrom these properties it follows that for all k, and in particular,.\nHence the k-th order Taylor polynomial of f at 0 and its remainder term in the Lagrange form are given by\nwhere \\xi is some number between 0 and x.\nSince ex is increasing by, we can simply use e^x \\leq 1 for to estimate the remainder on the subinterval [-1,0].\nTo obtain an upper bound for the remainder on [0,1], we use the property e^\\xi 0 and a sequence of coefficients ck ∈ **R** such that (a − r, a + r) ⊂ I and\nIn general, the [radius of convergence](https://bliptext.com/articles/power-series) of a power series can be computed from the [Cauchy–Hadamard formula](https://bliptext.com/articles/cauchy-hadamard-theorem)\nThis result is based on comparison with a [geometric series](https://bliptext.com/articles/geometric-series), and the same method shows that if the power series based on a converges for some b ∈ **R**, it must converge [uniformly](https://bliptext.com/articles/uniform-convergence) on the [closed interval](https://bliptext.com/articles/closed-interval), where.\nHere only the convergence of the power series is considered, and it might well be that (a − R,a + R) extends beyond the domain I of f.\nThe Taylor polynomials of the real analytic function f at a are simply the finite truncations\nof its locally defining power series, and the corresponding remainder terms are locally given by the analytic functions\nHere the functions\nare also analytic, since their defining power series have the same radius of convergence as the original series.\nAssuming that [a − r, a + r] ⊂ I and r < R, all these series converge uniformly on (a − r, a + r).\nNaturally, in the case of analytic functions one can estimate the remainder term R_k(x) by the tail of the sequence of the derivatives f′(a) at the center of the expansion, but using [complex analysis](https://bliptext.com/articles/complex-analysis) also another possibility arises, which is described [below](https://bliptext.com/articles/taylor-s-theorem).\n\n## Taylor's theorem and convergence of Taylor series\nThe Taylor series of f will converge in some interval in which all its derivatives are bounded and do not grow too fast as k goes to infinity.\n(However, even if the Taylor series converges, it might not converge to f, as explained below; f is then said to be non-[analytic](https://bliptext.com/articles/analytic-function).)\nOne might think of the Taylor series\nof an infinitely many times differentiable function f : ****R**** → ****R**** as its \"infinite order Taylor polynomial\" at a.\nNow the [estimates for the remainder](https://bliptext.com/articles/taylor-s-theorem) imply that if, for any r, the derivatives of f are known to be bounded over (a − r, a + r), then for any order k and for any r > 0 there exists a constant Mk,r > 0 such that\nfor every x ∈ (a − r,a + r).\nSometimes the constants Mk,r can be chosen in such way that Mk,r is bounded above, for fixed r and all k.\nThen the Taylor series of f [converges uniformly](https://bliptext.com/articles/uniform-convergence) to some analytic function\n(One also gets convergence even if Mk,r is not bounded above as long as it grows slowly enough.)\nThe limit function Tf is by definition always analytic, but it is not necessarily equal to the original function f, even if f is infinitely differentiable.\nIn this case, we say f is a [non-analytic smooth function](https://bliptext.com/articles/non-analytic-smooth-function), for example a [flat function](https://bliptext.com/articles/flat-function):\nUsing the [chain rule](https://bliptext.com/articles/chain-rule) repeatedly by [mathematical induction](https://bliptext.com/articles/mathematical-induction), one shows that for any order k,\nfor some polynomial pk of degree 2(k − 1).\nThe function tends to zero faster than any polynomial as x \\to 0, so f is infinitely many times differentiable and f(0) = 0 for every positive integer k.\nThe above results all hold in this case:\nHowever, as k increases for fixed r, the value of Mk,r grows more quickly than rk, and the error does not go to zero.\n\n## Taylor's theorem in complex analysis\nTaylor's theorem generalizes to functions f : ******C****** → ******C****** which are [complex differentiable](https://bliptext.com/articles/complex-differentiable) in an open subset U ⊂ ******C****** of the [complex plane](https://bliptext.com/articles/complex-plane).\nHowever, its usefulness is dwarfed by other general theorems in [complex analysis](https://bliptext.com/articles/complex-analysis).\nNamely, stronger versions of related results can be deduced for [complex differentiable](https://bliptext.com/articles/complex-differentiable) functions f : U → **C** using **C**auchy's integral formula as follows.\nLet r > 0 such that the [closed disk](https://bliptext.com/articles/closed-disk) B(z, r) ∪ S(z, r) is contained in U.\nThen Cauchy's integral formula with a positive parametrization γ(t) = z + reit of the circle S(z, r) with gives\nHere all the integrands are continuous on the [circle](https://bliptext.com/articles/circle) S(z, r), which justifies differentiation under the integral sign.\nIn particular, if f is once [complex differentiable](https://bliptext.com/articles/complex-differentiable) on the open set U, then it is actually infinitely many times [complex differentiable](https://bliptext.com/articles/complex-differentiable) on U.\nOne also obtains [Cauchy's estimate](https://bliptext.com/articles/cauchy-s-estimate)\nfor any z ∈ U and r > 0 such that B(z, r) ∪ S(c, r) ⊂ U.\nThe estimate implies that the [complex](https://bliptext.com/articles/complex-number) [Taylor series](https://bliptext.com/articles/taylor-series)\nof f converges uniformly on any [open disk](https://bliptext.com/articles/open-disk) with into some function Tf.\nFurthermore, using the [contour integral](https://bliptext.com/articles/contour-integral) formulas for the derivatives f(c),\nso any [complex differentiable](https://bliptext.com/articles/complex-derivative) function f in an open set U ⊂ **C** is in fact [complex analytic](https://bliptext.com/articles/complex-analytic).\nAll that is said for real analytic functions [here](https://bliptext.com/articles/taylor-s-theorem) holds also for complex analytic functions with the open interval I replaced by an open subset U ∈ **C** and a-centered intervals (a − r, a + r) replaced by c-centered disks B(c, r).\nIn particular, the Taylor expansion holds in the form\nwhere the remainder term Rk is complex analytic.\nMethods of complex analysis provide some powerful results regarding Taylor expansions.\nFor example, using Cauchy's integral formula for any positively oriented [Jordan curve](https://bliptext.com/articles/jordan-curve) \\gamma which parametrizes the boundary of a region W \\subset U, one obtains expressions for the derivatives f(c) as above, and modifying slightly the computation for Tf(z) = f(z), one arrives at the exact formula\nThe important feature here is that the quality of the approximation by a Taylor polynomial on the region W \\subset U is dominated by the values of the function f itself on the boundary.\nSimilarly, applying Cauchy's estimates to the series expression for the remainder, one obtains the uniform estimates\n\n## Example\nThe function\nis [real analytic](https://bliptext.com/articles/analytic-function), that is, locally determined by its Taylor series.\nThis function was plotted [above](https://bliptext.com/articles/taylor-s-theorem) to illustrate the fact that some elementary functions cannot be approximated by Taylor polynomials in neighborhoods of the center of expansion which are too large.\nThis kind of behavior is easily understood in the framework of complex analysis.\nNamely, the function f extends into a [meromorphic function](https://bliptext.com/articles/meromorphic-function)\non the compactified complex plane.\nIt has simple poles at z=i and z=-i, and it is analytic elsewhere.\nNow its Taylor series centered at z0 converges on any disc B(z0, r) with r < |z − z0|, where the same Taylor series converges at z ∈ **C**.\nTherefore, Taylor series of f centered at 0 converges on B(0, 1) and it does not converge for any z ∈ **C** with |z| > 1 due to the poles at i and −i.\nFor the same reason the Taylor series of f centered at 1 converges on and does not converge for any z ∈ **C** with.\n# [Complex plot of . Modulus is shown by elevation and argument by coloring: cyan = 0, blue =, violet =, red = \\pi, yellow = , green =. | upload.wikimedia.org/wikipedia/commons/b/ba/Function///with///two///poles.png]\n\n# Generalizations of Taylor's theorem\n\n## Higher-order differentiability\nA function\nf: Rn → R\nis [differentiable](https://bliptext.com/articles/derivative) at\na ∈ Rn\n[if and only if](https://bliptext.com/articles/if-and-only-if) there exists a [linear functional](https://bliptext.com/articles/linear-functional)\nL : Rn → R\nand a function\nh : Rn → R\nsuch that\nIf this is the case, then is the (uniquely defined) [differential](https://bliptext.com/articles/differential-of-a-function) of\nf\nat the point\na\n. Furthermore, then the [partial derivatives](https://bliptext.com/articles/partial-derivatives) of\nf\nexist at\na\nand the differential of\nf\nat\na\nis given by\nIntroduce the [multi-index notation](https://bliptext.com/articles/multi-index-notation)\nfor\nα ∈ Nn\nand\nx ∈ Rn\n. If all the k-th order [partial derivatives](https://bliptext.com/articles/partial-derivatives) of\nf : Rn → R\nare continuous at\na ∈ Rn\n, then by [Clairaut's theorem](https://bliptext.com/articles/symmetry-of-second-derivatives), one can change the order of mixed derivatives at\na\n, so the short-hand notation\nfor the higher order [partial derivatives](https://bliptext.com/articles/partial-derivatives) is justified in this situation.\nThe same is true if all the (\nk − 1\n)-th order partial derivatives of\nf\nexist in some neighborhood of\na\nand are differentiable at\na\n. Then we say that\nf\nis\nk\n'''times differentiable at the point\na\n'''.\n\n## Taylor's theorem for multivariate functions\nUsing notations of the preceding section, one has the following theorem.\nIf the function\nf : Rn → R\nis\nk + 1\ntimes [continuously differentiable](https://bliptext.com/articles/continuously-differentiable) in a [closed ball](https://bliptext.com/articles/closed-ball) for some r > 0, then one can derive an exact formula for the remainder in terms of (\nk+1\n)-th order [partial derivatives](https://bliptext.com/articles/partial-derivatives) of f in this neighborhood. Namely,\nIn this case, due to the [continuity](https://bliptext.com/articles/continuous-function) of (\nk+1\n)-th order [partial derivatives](https://bliptext.com/articles/partial-derivative) in the [compact set](https://bliptext.com/articles/compact-set)\nB\n, one immediately obtains the uniform estimates\n\n## Example in two dimensions\nFor example, the third-order Taylor polynomial of a smooth function is, denoting ,\n\n# Proofs\n\n## Proof for Taylor's theorem in one real variable\nLet\nwhere, as in the statement of Taylor's theorem,\nIt is sufficient to show that\nThe proof here is based on repeated application of [L'Hôpital's rule](https://bliptext.com/articles/l-h-pital-s-rule).\nNote that, for each,.\nHence each of the first k-1 derivatives of the numerator in h_k(x) vanishes at x=a, and the same is true of the denominator.\nAlso, since the condition that the function f be k times differentiable at a point requires differentiability up to order k-1 in a neighborhood of said point (this is true, because differentiability requires a function to be defined in a whole neighborhood of a point), the numerator and its k-2 derivatives are differentiable in a neighborhood of a. Clearly, the denominator also satisfies said condition, and additionally, doesn't vanish unless x=a, therefore all conditions necessary for L'Hôpital's rule are fulfilled, and its use is justified.\nSo\nwhere the second-to-last equality follows by the definition of the derivative at x=a.\n\n## Alternate proof for Taylor's theorem in one real variable\nLet f(x) be any real-valued continuous function to be approximated by the Taylor polynomial.\nStep 1: Let F and G be functions.\nSet F and G to be\nStep 2: Properties of F and G:\nSimilarly,\nStep 3: Use Cauchy Mean Value Theorem\nLet f_{1} and g_{1} be continuous functions on [a, b].\nSince a < x < b so we can work with the interval [a, x].\nLet f_{1} and g_{1} be differentiable on (a, x).\nAssume for all.\nThen there exists such that\nNote: in (a, b) and so\nfor some.\nThis can also be performed for (a, c_{1}):\nfor some.\nThis can be continued to c_{n}.\nThis gives a partition in (a, b):\nwith\nSet c = c_{n}:\nStep 4: Substitute back\nBy the Power Rule, repeated derivatives of (x - a)^{n},, so:\nThis leads to:\nBy rearranging, we get:\nor because c_{n} = a eventually:\n\n## Derivation for the mean value forms of the remainder\nLet G be any real-valued function, continuous on the closed interval between a and x and differentiable with a non-vanishing derivative on the open interval between a and x, and define\nFor t \\in [a,x].\nThen, by [Cauchy's mean value theorem](https://bliptext.com/articles/mean-value-theorem),\nfor some \\xi on the open interval between a and x. Note that here the numerator is exactly the remainder of the Taylor polynomial for y=f(x).\nCompute\nplug it into and rearrange terms to find that\nThis is the form of the remainder term mentioned after the actual statement of Taylor's theorem with remainder in the mean value form.\nThe Lagrange form of the remainder is found by choosing and the Cauchy form by choosing G(t) = t-a.\n**Remark.** Using this method one can also recover the integral form of the remainder by choosing\nbut the requirements for f needed for the use of mean value theorem are too strong, if one aims to prove the claim in the case that f is only [absolutely continuous](https://bliptext.com/articles/absolutely-continuous).\nHowever, if one uses [Riemann integral](https://bliptext.com/articles/riemann-integral) instead of [Lebesgue integral](https://bliptext.com/articles/lebesgue-integral), the assumptions cannot be weakened.\n\n## Derivation for the integral form of the remainder\nDue to the [absolute continuity](https://bliptext.com/articles/absolutely-continuous) of f^{(k)} on the [closed interval](https://bliptext.com/articles/closed-interval) between a and x, its derivative f^{(k+1)} exists as an L^1-function, and we can use the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) and [integration by parts](https://bliptext.com/articles/integration-by-parts).\nThis same proof applies for the [Riemann integral](https://bliptext.com/articles/riemann-integral) assuming that f^{(k)} is [continuous](https://bliptext.com/articles/continuous-function) on the closed interval and [differentiable](https://bliptext.com/articles/differentiable-function) on the [open interval](https://bliptext.com/articles/open-interval) between a and x, and this leads to the same result than using the mean value theorem.\nThe [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) states that\nNow we can [integrate by parts](https://bliptext.com/articles/integration-by-parts) and use the fundamental theorem of calculus again to see that\nwhich is exactly Taylor's theorem with remainder in the integral form in the case k=1.\nThe general statement is proved using [induction](https://bliptext.com/articles/mathematical-induction).\nSuppose that\nIntegrating the remainder term by parts we arrive at\nSubstituting this into the formula in shows that if it holds for the value k, it must also hold for the value k+1.\nTherefore, since it holds for k=1, it must hold for every positive integer k.\n\n## Derivation for the remainder of multivariate Taylor polynomials\nWe prove the special case, where has continuous partial derivatives up to the order k+1 in some closed ball B with center.\nThe strategy of the proof is to apply the one-variable case of Taylor's theorem to the restriction of f to the line segment adjoining and.\nParametrize the line segment between and by We apply the one-variable version of Taylor's theorem to the function :\nApplying the [chain rule](https://bliptext.com/articles/chain-rule) for several variables gives\nwhere is the [multinomial coefficient](https://bliptext.com/articles/multinomial-coefficient).\nSince, we get:\n\n# Footnotes","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:06:32.159Z","updated_at":"2025-01-25T18:06:32.159Z","search_vector":"'taylor':1 'theorem':3","search_content":" \n \n\n# \nIn [calculus](https://bliptext.com/articles/calculus), **Taylor's theorem** gives an approximation of a k-times [differentiable function](https://bliptext.com/articles/differentiable-function) around a given point by a [polynomial](https://bliptext.com/articles/polynomial) of degree k, called the k-th-order **Taylor [polynomial](https://bliptext.com/articles/polynomial)**.\nFor a [smooth function](https://bliptext.com/articles/smooth-function), the Taylor polynomial is the truncation at the order k of the [Taylor series](https://bliptext.com/articles/taylor-series) of the function.\nThe first-order Taylor polynomial is the [linear approximation](https://bliptext.com/articles/linear-approximation) of the function, and the second-order Taylor polynomial is often referred to as the **quadratic approximation**.\nThere are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.\nTaylor's theorem is named after the mathematician [Brook Taylor](https://bliptext.com/articles/brook-taylor), who stated a version of it in 1715, although an earlier version of the result was already mentioned in [1671](https://bliptext.com/articles/1671-in-science) by [James Gregory](https://bliptext.com/articles/james-gregory-astronomer-and-mathematician).\nTaylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in [mathematical analysis](https://bliptext.com/articles/mathematical-analysis).\nIt gives simple arithmetic formulas to accurately compute values of many [transcendental functions](https://bliptext.com/articles/transcendental-function) such as the [exponential function](https://bliptext.com/articles/exponential-function) and [trigonometric functions](https://bliptext.com/articles/trigonometric-function).\nIt is the starting point of the study of [analytic functions](https://bliptext.com/articles/analytic-function), and is fundamental in various areas of mathematics, as well as in [numerical analysis](https://bliptext.com/articles/numerical-analysis) and [mathematical physics](https://bliptext.com/articles/mathematical-physics).\nTaylor's theorem also generalizes to [multivariate](https://bliptext.com/articles/multivariate-function) and [vector valued](https://bliptext.com/articles/vector-valued-function) functions.\nIt provided the mathematical basis for some landmark early computing machines: [Charles Babbage](https://bliptext.com/articles/charles-babbage)'s [Difference Engine](https://bliptext.com/articles/difference-engine) calculated sines, cosines, logarithms, and other transcendental functions by numerically integrating the first 7 terms of their Taylor series.\n# [300px | upload.wikimedia.org/wikipedia/commons/6/64/Taylorspolynomialexbig.svg]\n\n# Motivation\nIf a real-valued [function](https://bliptext.com/articles/function-mathematics) f(x) is [differentiable](https://bliptext.com/articles/derivative) at the point x=a, then it has a [linear approximation](https://bliptext.com/articles/linear-approximation) near this point.\nThis means that there exists a function h1(x) such that\nHere\nis the linear approximation of f(x) for x near the point a, whose graph y=P_1(x) is the [tangent line](https://bliptext.com/articles/tangent-line) to the graph y=f(x) at x = a. The error in the approximation is:\nAs x tends to a, this error goes to zero much faster than, making a useful approximation.\nFor a better approximation to f(x), we can fit a [quadratic polynomial](https://bliptext.com/articles/quadratic-polynomial) instead of a linear function:\nInstead of just matching one derivative of f(x) at x=a, this polynomial has the same first and second derivatives, as is evident upon differentiation.\nTaylor's theorem ensures that the quadratic approximation is, in a sufficiently small neighborhood of x=a, more accurate than the linear approximation.\nSpecifically,\nHere the error in the approximation is\nwhich, given the limiting behavior of h_2, goes to zero faster than (x - a)^2 as x tends to a.\nSimilarly, we might get still better approximations to f if we use [polynomials](https://bliptext.com/articles/polynomial) of higher degree, since then we can match even more derivatives with f at the selected base point.\nIn general, the error in approximating a function by a polynomial of degree k will go to zero much faster than (x-a)^k as x tends to a.\nHowever, there are functions, even infinitely differentiable ones, for which increasing the degree of the approximating polynomial does not increase the accuracy of approximation: we say such a function fails to be [analytic](https://bliptext.com/articles/analytic-function) at x = a: it is not (locally) determined by its derivatives at this point.\nTaylor's theorem is of asymptotic nature: it only tells us that the error R_k in an [approximation](https://bliptext.com/articles/approximation) by a k-th order Taylor [polynomial](https://bliptext.com/articles/polynomial) Pk tends to zero faster than any nonzero k-th degree [polynomial](https://bliptext.com/articles/polynomial) as x \\to a. It does not tell us how large the error is in any concrete [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) of the center of expansion, but for this purpose there are explicit formulas for the remainder term (given below) which are valid under some additional regularity assumptions on f.\nThese enhanced versions of Taylor's theorem typically lead to [uniform estimates](https://bliptext.com/articles/uniform-convergence) for the approximation error in a small neighborhood of the center of expansion, but the estimates do not necessarily hold for neighborhoods which are too large, even if the function f is [analytic](https://bliptext.com/articles/analytic-function).\nIn that situation one may have to select several Taylor polynomials with different centers of expansion to have reliable Taylor-approximations of the original function (see animation on the right.)\nThere are several ways we might use the remainder term:\n# [Taylor's theorem | upload.wikimedia.org/wikipedia/commons/6/6e/E%5Ex///with///linear///approximation.png]\n# [Taylor's theorem | upload.wikimedia.org/wikipedia/commons/6/6b/E%5Ex///with///quadratic///approximation///corrected.png]\n\n# Taylor's theorem in one real variable\n\n## Statement of the theorem\nThe precise statement of the most basic version of Taylor's theorem is as follows:\nThe polynomial appearing in Taylor's theorem is the **-th order Taylor polynomial**\nof the function f at the point a.\nThe Taylor polynomial is the unique \"asymptotic best fit\" polynomial in the sense that if there exists a function hk : R → R and a k-th order polynomial p such that\nthen p = Pk.\nTaylor's theorem describes the asymptotic behavior of the **remainder term**\nwhich is the [approximation error](https://bliptext.com/articles/approximation-error) when approximating f with its Taylor polynomial.\nUsing the [little-o notation](https://bliptext.com/articles/little-o-notation), the statement in Taylor's theorem reads as\n\n## Explicit formulas for the remainder\nUnder stronger regularity assumptions on f there are several precise formulas for the remainder term Rk of the Taylor polynomial, the most common ones being the following.\nThese refinements of Taylor's theorem are usually proved using the [mean value theorem](https://bliptext.com/articles/mean-value-theorem), whence the name.\nAdditionally, notice that this is precisely the [mean value theorem](https://bliptext.com/articles/mean-value-theorem) when k=0.\nAlso other similar expressions can be found.\nFor example, if G(t) is continuous on the closed interval and differentiable with a non-vanishing derivative on the open interval between a and x, then\nfor some number \\xi between a and x. This version covers the Lagrange and Cauchy forms of the remainder as special cases, and is proved below using [Cauchy's mean value theorem](https://bliptext.com/articles/mean-value-theorem).\nThe Lagrange form is obtained by taking and the Cauchy form is obtained by taking G(t)=t-a.\nThe statement for the integral form of the remainder is more advanced than the previous ones, and requires understanding of [Lebesgue integration theory](https://bliptext.com/articles/lebesgue-integral) for the full generality.\nHowever, it holds also in the sense of [Riemann integral](https://bliptext.com/articles/riemann-integral) provided the (k + 1)th derivative of f is continuous on the closed interval [a,x].\nDue to the [absolute continuity](https://bliptext.com/articles/absolutely-continuous) of f on the [closed interval](https://bliptext.com/articles/closed-interval) between a and x, its derivative f exists as an L-function, and the result can be [proven](https://bliptext.com/articles/) by a formal calculation using the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) and [integration by parts](https://bliptext.com/articles/integration-by-parts).\n\n## Estimates for the remainder\nIt is often useful in practice to be able to estimate the remainder term appearing in the Taylor approximation, rather than having an exact formula for it.\nSuppose that f is (k + 1)-times continuously differentiable in an interval I containing a.\nSuppose that there are real constants q and Q such that\nthroughout I.\nThen the remainder term satisfies the inequality\nif x > a, and a similar estimate if x 0, then\nfor all x∈(a − r,a + r).\nThe second inequality is called a [uniform estimate](https://bliptext.com/articles/uniform-convergence), because it holds uniformly for all x on the interval (a − r,a + r).\n\n## Example\nSuppose that we wish to find the approximate value of the function f(x)=e^x on the interval [-1,1] while ensuring that the error in the approximation is no more than 10−5.\nIn this example we pretend that we only know the following properties of the exponential function:\nFrom these properties it follows that for all k, and in particular,.\nHence the k-th order Taylor polynomial of f at 0 and its remainder term in the Lagrange form are given by\nwhere \\xi is some number between 0 and x.\nSince ex is increasing by, we can simply use e^x \\leq 1 for to estimate the remainder on the subinterval [-1,0].\nTo obtain an upper bound for the remainder on [0,1], we use the property e^\\xi 0 and a sequence of coefficients ck ∈ **R** such that (a − r, a + r) ⊂ I and\nIn general, the [radius of convergence](https://bliptext.com/articles/power-series) of a power series can be computed from the [Cauchy–Hadamard formula](https://bliptext.com/articles/cauchy-hadamard-theorem)\nThis result is based on comparison with a [geometric series](https://bliptext.com/articles/geometric-series), and the same method shows that if the power series based on a converges for some b ∈ **R**, it must converge [uniformly](https://bliptext.com/articles/uniform-convergence) on the [closed interval](https://bliptext.com/articles/closed-interval), where.\nHere only the convergence of the power series is considered, and it might well be that (a − R,a + R) extends beyond the domain I of f.\nThe Taylor polynomials of the real analytic function f at a are simply the finite truncations\nof its locally defining power series, and the corresponding remainder terms are locally given by the analytic functions\nHere the functions\nare also analytic, since their defining power series have the same radius of convergence as the original series.\nAssuming that [a − r, a + r] ⊂ I and r 0 there exists a constant Mk,r > 0 such that\nfor every x ∈ (a − r,a + r).\nSometimes the constants Mk,r can be chosen in such way that Mk,r is bounded above, for fixed r and all k.\nThen the Taylor series of f [converges uniformly](https://bliptext.com/articles/uniform-convergence) to some analytic function\n(One also gets convergence even if Mk,r is not bounded above as long as it grows slowly enough.)\nThe limit function Tf is by definition always analytic, but it is not necessarily equal to the original function f, even if f is infinitely differentiable.\nIn this case, we say f is a [non-analytic smooth function](https://bliptext.com/articles/non-analytic-smooth-function), for example a [flat function](https://bliptext.com/articles/flat-function):\nUsing the [chain rule](https://bliptext.com/articles/chain-rule) repeatedly by [mathematical induction](https://bliptext.com/articles/mathematical-induction), one shows that for any order k,\nfor some polynomial pk of degree 2(k − 1).\nThe function tends to zero faster than any polynomial as x \\to 0, so f is infinitely many times differentiable and f(0) = 0 for every positive integer k.\nThe above results all hold in this case:\nHowever, as k increases for fixed r, the value of Mk,r grows more quickly than rk, and the error does not go to zero.\n\n## Taylor's theorem in complex analysis\nTaylor's theorem generalizes to functions f : ******C****** → ******C****** which are [complex differentiable](https://bliptext.com/articles/complex-differentiable) in an open subset U ⊂ ******C****** of the [complex plane](https://bliptext.com/articles/complex-plane).\nHowever, its usefulness is dwarfed by other general theorems in [complex analysis](https://bliptext.com/articles/complex-analysis).\nNamely, stronger versions of related results can be deduced for [complex differentiable](https://bliptext.com/articles/complex-differentiable) functions f : U → **C** using **C**auchy's integral formula as follows.\nLet r > 0 such that the [closed disk](https://bliptext.com/articles/closed-disk) B(z, r) ∪ S(z, r) is contained in U.\nThen Cauchy's integral formula with a positive parametrization γ(t) = z + reit of the circle S(z, r) with gives\nHere all the integrands are continuous on the [circle](https://bliptext.com/articles/circle) S(z, r), which justifies differentiation under the integral sign.\nIn particular, if f is once [[complex differentiable](https://bliptext.com/articles/complex-differentiable)](https://bliptext.com/articles/complex-differentiable) on the open set U, then it is actually infinitely many times [[complex differentiable](https://bliptext.com/articles/complex-differentiable)](https://bliptext.com/articles/complex-differentiable) on U.\nOne also obtains [Cauchy's estimate](https://bliptext.com/articles/cauchy-s-estimate)\nfor any z ∈ U and r > 0 such that B(z, r) ∪ S(c, r) ⊂ U.\nThe estimate implies that the [complex](https://bliptext.com/articles/complex-number) [Taylor series](https://bliptext.com/articles/taylor-series)\nof f converges uniformly on any [open disk](https://bliptext.com/articles/open-disk) with into some function Tf.\nFurthermore, using the [contour integral](https://bliptext.com/articles/contour-integral) formulas for the derivatives f(c),\nso any [complex differentiable](https://bliptext.com/articles/complex-derivative) function f in an open set U ⊂ **C** is in fact [complex analytic](https://bliptext.com/articles/complex-analytic).\nAll that is said for real analytic functions [here](https://bliptext.com/articles/taylor-s-theorem) holds also for complex analytic functions with the open interval I replaced by an open subset U ∈ **C** and a-centered intervals (a − r, a + r) replaced by c-centered disks B(c, r).\nIn particular, the Taylor expansion holds in the form\nwhere the remainder term Rk is complex analytic.\nMethods of complex analysis provide some powerful results regarding Taylor expansions.\nFor example, using Cauchy's integral formula for any positively oriented [Jordan curve](https://bliptext.com/articles/jordan-curve) \\gamma which parametrizes the boundary of a region W \\subset U, one obtains expressions for the derivatives f(c) as above, and modifying slightly the computation for Tf(z) = f(z), one arrives at the exact formula\nThe important feature here is that the quality of the approximation by a Taylor polynomial on the region W \\subset U is dominated by the values of the function f itself on the boundary.\nSimilarly, applying Cauchy's estimates to the series expression for the remainder, one obtains the uniform estimates\n\n## Example\nThe function\nis [real analytic](https://bliptext.com/articles/analytic-function), that is, locally determined by its Taylor series.\nThis function was plotted [above](https://bliptext.com/articles/taylor-s-theorem) to illustrate the fact that some elementary functions cannot be approximated by Taylor polynomials in neighborhoods of the center of expansion which are too large.\nThis kind of behavior is easily understood in the framework of complex analysis.\nNamely, the function f extends into a [meromorphic function](https://bliptext.com/articles/meromorphic-function)\non the compactified complex plane.\nIt has simple poles at z=i and z=-i, and it is analytic elsewhere.\nNow its Taylor series centered at z0 converges on any disc B(z0, r) with r 1 due to the poles at i and −i.\nFor the same reason the Taylor series of f centered at 1 converges on and does not converge for any z ∈ **C** with.\n# [Complex plot of . Modulus is shown by elevation and argument by coloring: cyan = 0, blue =, violet =, red = \\pi, yellow = , green =. | upload.wikimedia.org/wikipedia/commons/b/ba/Function///with///two///poles.png]\n\n# Generalizations of Taylor's theorem\n\n## Higher-order differentiability\nA function\nf: Rn → R\nis [differentiable](https://bliptext.com/articles/derivative) at\na ∈ Rn\n[if and only if](https://bliptext.com/articles/if-and-only-if) there exists a [linear functional](https://bliptext.com/articles/linear-functional)\nL : Rn → R\nand a function\nh : Rn → R\nsuch that\nIf this is the case, then is the (uniquely defined) [differential](https://bliptext.com/articles/differential-of-a-function) of\nf\nat the point\na\n. Furthermore, then the [partial derivatives](https://bliptext.com/articles/partial-derivatives) of\nf\nexist at\na\nand the differential of\nf\nat\na\nis given by\nIntroduce the [multi-index notation](https://bliptext.com/articles/multi-index-notation)\nfor\nα ∈ Nn\nand\nx ∈ Rn\n. If all the k-th order [partial derivatives](https://bliptext.com/articles/partial-derivatives) of\nf : Rn → R\nare continuous at\na ∈ Rn\n, then by [Clairaut's theorem](https://bliptext.com/articles/symmetry-of-second-derivatives), one can change the order of mixed derivatives at\na\n, so the short-hand notation\nfor the higher order [partial derivatives](https://bliptext.com/articles/partial-derivatives) is justified in this situation.\nThe same is true if all the (\nk − 1\n)-th order partial derivatives of\nf\nexist in some neighborhood of\na\nand are differentiable at\na\n. Then we say that\nf\nis\nk\n'''times differentiable at the point\na\n'''.\n\n## Taylor's theorem for multivariate functions\nUsing notations of the preceding section, one has the following theorem.\nIf the function\nf : Rn → R\nis\nk + 1\ntimes [continuously differentiable](https://bliptext.com/articles/continuously-differentiable) in a [closed ball](https://bliptext.com/articles/closed-ball) for some r > 0, then one can derive an exact formula for the remainder in terms of (\nk+1\n)-th order [partial derivatives](https://bliptext.com/articles/partial-derivatives) of f in this neighborhood. Namely,\nIn this case, due to the [continuity](https://bliptext.com/articles/continuous-function) of (\nk+1\n)-th order [partial derivatives](https://bliptext.com/articles/partial-derivative) in the [compact set](https://bliptext.com/articles/compact-set)\nB\n, one immediately obtains the uniform estimates\n\n## Example in two dimensions\nFor example, the third-order Taylor polynomial of a smooth function is, denoting ,\n\n# Proofs\n\n## Proof for Taylor's theorem in one real variable\nLet\nwhere, as in the statement of Taylor's theorem,\nIt is sufficient to show that\nThe proof here is based on repeated application of [L'Hôpital's rule](https://bliptext.com/articles/l-h-pital-s-rule).\nNote that, for each,.\nHence each of the first k-1 derivatives of the numerator in h_k(x) vanishes at x=a, and the same is true of the denominator.\nAlso, since the condition that the function f be k times differentiable at a point requires differentiability up to order k-1 in a neighborhood of said point (this is true, because differentiability requires a function to be defined in a whole neighborhood of a point), the numerator and its k-2 derivatives are differentiable in a neighborhood of a. Clearly, the denominator also satisfies said condition, and additionally, doesn't vanish unless x=a, therefore all conditions necessary for L'Hôpital's rule are fulfilled, and its use is justified.\nSo\nwhere the second-to-last equality follows by the definition of the derivative at x=a.\n\n## Alternate proof for Taylor's theorem in one real variable\nLet f(x) be any real-valued continuous function to be approximated by the Taylor polynomial.\nStep 1: Let F and G be functions.\nSet F and G to be\nStep 2: Properties of F and G:\nSimilarly,\nStep 3: Use Cauchy Mean Value Theorem\nLet f_{1} and g_{1} be continuous functions on [a, b].\nSince a < x < b so we can work with the interval [a, x].\nLet f_{1} and g_{1} be differentiable on (a, x).\nAssume for all.\nThen there exists such that\nNote: in (a, b) and so\nfor some.\nThis can also be performed for (a, c_{1}):\nfor some.\nThis can be continued to c_{n}.\nThis gives a partition in (a, b):\nwith\nSet c = c_{n}:\nStep 4: Substitute back\nBy the Power Rule, repeated derivatives of (x - a)^{n},, so:\nThis leads to:\nBy rearranging, we get:\nor because c_{n} = a eventually:\n\n## Derivation for the mean value forms of the remainder\nLet G be any real-valued function, continuous on the closed interval between a and x and differentiable with a non-vanishing derivative on the open interval between a and x, and define\nFor t \\in [a,x].\nThen, by [Cauchy's mean value theorem](https://bliptext.com/articles/mean-value-theorem),\nfor some \\xi on the open interval between a and x. Note that here the numerator is exactly the remainder of the Taylor polynomial for y=f(x).\nCompute\nplug it into and rearrange terms to find that\nThis is the form of the remainder term mentioned after the actual statement of Taylor's theorem with remainder in the mean value form.\nThe Lagrange form of the remainder is found by choosing and the Cauchy form by choosing G(t) = t-a.\n**Remark.** Using this method one can also recover the integral form of the remainder by choosing\nbut the requirements for f needed for the use of mean value theorem are too strong, if one aims to prove the claim in the case that f is only [absolutely continuous](https://bliptext.com/articles/absolutely-continuous).\nHowever, if one uses [Riemann integral](https://bliptext.com/articles/riemann-integral) instead of [Lebesgue integral](https://bliptext.com/articles/lebesgue-integral), the assumptions cannot be weakened.\n\n## Derivation for the integral form of the remainder\nDue to the [absolute continuity](https://bliptext.com/articles/absolutely-continuous) of f^{(k)} on the [closed interval](https://bliptext.com/articles/closed-interval) between a and x, its derivative f^{(k+1)} exists as an L^1-function, and we can use the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) and [integration by parts](https://bliptext.com/articles/integration-by-parts).\nThis same proof applies for the [Riemann integral](https://bliptext.com/articles/riemann-integral) assuming that f^{(k)} is [continuous](https://bliptext.com/articles/continuous-function) on the closed interval and [differentiable](https://bliptext.com/articles/differentiable-function) on the [open interval](https://bliptext.com/articles/open-interval) between a and x, and this leads to the same result than using the mean value theorem.\nThe [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) states that\nNow we can [integrate by parts](https://bliptext.com/articles/integration-by-parts) and use the fundamental theorem of calculus again to see that\nwhich is exactly Taylor's theorem with remainder in the integral form in the case k=1.\nThe general statement is proved using [induction](https://bliptext.com/articles/mathematical-induction).\nSuppose that\nIntegrating the remainder term by parts we arrive at\nSubstituting this into the formula in shows that if it holds for the value k, it must also hold for the value k+1.\nTherefore, since it holds for k=1, it must hold for every positive integer k.\n\n## Derivation for the remainder of multivariate Taylor polynomials\nWe prove the special case, where has continuous partial derivatives up to the order k+1 in some closed ball B with center.\nThe strategy of the proof is to apply the one-variable case of Taylor's theorem to the restriction of f to the line segment adjoining and.\nParametrize the line segment between and by We apply the one-variable version of Taylor's theorem to the function :\nApplying the [chain rule](https://bliptext.com/articles/chain-rule) for several variables gives\nwhere is the [multinomial coefficient](https://bliptext.com/articles/multinomial-coefficient).\nSince, we get:\n\n# Footnotes","revisionCount":0},"hasMore":false,"nextCursor":null}

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