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 positive, strictly increasing and unbounded and the following limit exists:

:$lim\_\{n\; o\; infty\}\; frac\{a\_\{n+1\}-a\_n\}\{b\_\{n+1\}-b\_n\}=l.$

Then, the limit:

:$lim\_\{n\; o\; infty\}\; frac\{a\_n\}\{b\_n\}$

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.

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

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External links

* [http://planetmath.org/encyclopedia/ProofOfStolzCesaroTheorem.html Proof of Stolz-Cesàro theorem]