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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 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.

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

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