In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. It is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time. The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.

Statement of the theorem for the

*/∞

case Let and be two sequences of real numbers. Assume that is a strictly monotone and divergent sequence (i.e. strictly increasing and approaching + \infty, or strictly decreasing and approaching - \infty) and the following limit exists: Then, the limit

Statement of the theorem for the

0/0

case Let and be two sequences of real numbers. Assume now that (a_n)\to 0 and (b_n)\to 0 while is strictly decreasing. If then

Proofs

Proof of the theorem for the

*/∞

case Case 1: suppose (b_n) strictly increasing and divergent to +\infty, and. By hypothesis, we have that for all there exists \nu > 0 such that which is to say Since (b_n) is strictly increasing,, and the following holds Next we notice that thus, by applying the above inequality to each of the terms in the square brackets, we obtain Now, since as n\to\infty, there is an n_0>0 such that b_n>0 for all n>n_0, and we can divide the two inequalities by b_n for all The two sequences (which are only defined for n>n_0 as there could be an N\leq n_0 such that b_N=0) are infinitesimal since and the numerator is a constant number, hence for all there exists, such that therefore which concludes the proof. The case with (b_n) strictly decreasing and divergent to -\infty, and l<\infty is similar. Case 2: we assume (b_n) strictly increasing and divergent to +\infty, and l=+\infty. Proceeding as before, for all 2M > 0 there exists \nu > 0 such that for all n > \nu Again, by applying the above inequality to each of the terms inside the square brackets we obtain and The sequence defined by is infinitesimal, thus combining this inequality with the previous one we conclude The proofs of the other cases with (b_n) strictly increasing or decreasing and approaching +\infty or -\infty respectively and l=\pm\infty all proceed in this same way.

Proof of the theorem for the

0/0

case Case 1: we first consider the case with l < \infty and (b_n) strictly decreasing. This time, for each \nu > 0, we can write and for any \exist n_0 such that for all n>n_0 we have The two sequences are infinitesimal since by hypothesis as, thus for all there are such that thus, choosing \nu appropriately (which is to say, taking the limit with respect to \nu) we obtain which concludes the proof. Case 2: we assume l=+\infty and (b_n) strictly decreasing. For all 2M > 0 there exists n_0 > 0 such that for all n > n_0, Therefore, for each \nu > 0, The sequence converges to 0 (keeping n fixed). Hence and, choosing \nu conveniently, we conclude the proof

Applications and examples

The theorem concerning the ∞/∞ case has a few notable consequences which are useful in the computation of limits.

Arithmetic mean

Let (x_n) be a sequence of real numbers which converges to l, define then (b_n) is strictly increasing and diverges to +\infty. We compute therefore Given any sequence of real numbers, suppose that exists (finite or infinite), then

Geometric mean

Let (x_n) be a sequence of positive real numbers converging to l and define again we compute where we used the fact that the logarithm is continuous. Thus since the logarithm is both continuous and injective we can conclude that Given any sequence of (strictly) positive real numbers, suppose that exists (finite or infinite), then Suppose we are given a sequence and we are asked to compute defining y_0=1 and we obtain if we apply the property above This last form is usually the most useful to compute limits Given any sequence of (strictly) positive real numbers, suppose that exists (finite or infinite), then

Examples

Example 1

Example 2

where we used the representation of e as the limit of a sequence.

History

The ∞/∞ case is stated and proved on pages 173—175 of Stolz's 1885 book and also on page 54 of Cesàro's 1888 article. It appears as Problem 70 in Pólya and Szegő (1925).

The general form

Statement

The general form of the Stolz–Cesàro theorem is the following: If and are two sequences such that is monotone and unbounded, then:

Proof

Instead of proving the previous statement, we shall prove a slightly different one; first we introduce a notation: let be any sequence, its partial sum will be denoted by. The equivalent statement we shall prove is: Let be any two sequences of real numbers such that then

Proof of the equivalent statement

First we notice that: Therefore we need only to show that. If there is nothing to prove, hence we can assume L<+\infty (it can be either finite or -\infty). By definition of \limsup, for all l > L there is a natural number \nu>0 such that We can use this inequality so as to write Because b_n>0, we also have B_n>0 and we can divide by B_n to get Since as n\to+\infty, the sequence and we obtain By definition of least upper bound, this precisely means that and we are done.

Proof of the original statement

Now, take (a_n),(b_n) as in the statement of the general form of the Stolz-Cesàro theorem and define since (b_n) is strictly monotone (we can assume strictly increasing for example), \beta_n>0 for all n and since also, thus we can apply the theorem we have just proved to (and their partial sums ) which is exactly what we wanted to prove.