Stolz–Cesàro theorem Information & Stolz–Cesàro theorem Links at HealthHaven.com

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 $b n$ is strictly increasing and unbounded and the following limit exists:

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

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.

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