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# Stolz-Cesàro theorem

In mathematics, the Stolz-Cesàro theorem is a criterion for proving the convergence of a sequence.

Let $\left(a_n\right)_\left\{n geq 1\right\}$ and $\left(b_n\right)_\left\{n geq 1\right\}$ be two sequences of real numbers. Assume that $b_n$ is positive, strictly increasing and unbounded and the following limit exists:

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

Then, the limit:

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

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.

The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro.