Stolz-Cesàro theorem

In mathematics, the Stolz-Cesàro theorem 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 $b n$ is strictly increasing and unbounded and the following limit exists:

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

Then, the limit:

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

also exists and it is equal to $l .$

The Stolz-Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.

This article incorporates material from Stolz-Cesaro theorem on PlanetMath, which is licensed under the GFDL.

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