Stolz–Cesàro theorem

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In mathematics, the Stolz–Cesàro theorem, named after mathematicians Otto Stolz and Ernesto Cesàro, is a criterion for proving the convergence of a sequence.

Let (a_n)_{n \geq 1} and (b_n)_{n \geq 1} be two sequences of real numbers. Assume that (b_n)_{n \geq 1} is strictly monotone and divergent sequence (i.e. strictly increasing and approaches  + \infty or strictly decreasing and approaches  - \infty ) and the following limit exists:

 \lim_{n \to \infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}=\ell.\

Then, the limit

 \lim_{n \to \infty} \frac{a_n}{b_n}\

also exists and it is equal to .

The general form of the Stolz–Cesàro theorem is the following (see If  (a_n)_{n\geq 1} and  (b_n)_{n\geq 1} are two sequences such that (b_n)_{n \geq 1} is monotone and unbounded, then:

\liminf_{n\to\infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}\leq \liminf_{n\to\infty}\frac{a_n}{b_n}\leq\limsup_{n\to\infty}\frac{a_n}{b_n}\leq\limsup_{n\to\infty}\frac{a_{n+1}-a_n}{b_{n+1}-b_n}.

The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences. The ∞/∞ case is stated and proved on pages 173—175 of Stolz's 1885 book S and also on page 54 of Cesàro's 1888 article C. It appears as Problem 70 in Pólya and Szegö.


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This article incorporates material from Stolz-Cesaro theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.