# Stolz–Cesàro theorem

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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 $(b_n)_{n \geq 1}$ is strictly monotone and divergent sequence (i.e. strictly increasing and approaches $+ \infty$ or strictly decreasing and approaches $- \infty$) 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 .

The general form of the Stolz–Cesàro theorem is the following (see http://www.imomath.com/index.php?options=686): If $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ are two sequences such that $(b_n)_{n \geq 1}$ is monotone and unbounded, then:

$\liminf_{n\to\infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}\leq \liminf_{n\to\infty}\frac{a_n}{b_n}\leq\limsup_{n\to\infty}\frac{a_n}{b_n}\leq\limsup_{n\to\infty}\frac{a_{n+1}-a_n}{b_{n+1}-b_n}.$

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 ∞/∞ case is stated and proved on pages 173—175 of Stolz's 1885 book S and also on page 54 of Cesàro's 1888 article C. It appears as Problem 70 in Pólya and Szegö.