# Stolz–Cesàro theorem

In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time.

The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.

## Statement of the theorem for the ∙/∞ case

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}}}=\ell .\$ ## Statement of the theorem for the 0/0 case

Let $(a_{n})_{n\geq 1}$ and $(b_{n})_{n\geq 1}$ be two sequences of real numbers. Assume now that $(a_{n})\to 0$ and $(b_{n})\to 0$ while $(b_{n})_{n\geq 1}$ is strictly monotone. If

$\lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=\ell ,\$ then

$\lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=\ell .\$ ## History

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 Szegő.

## The general form

The general form of the Stolz–Cesàro theorem is the following: 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:

$\liminf _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}\leq \liminf _{n\to \infty }{\frac {a_{n}}{b_{n}}}\leq \limsup _{n\to \infty }{\frac {a_{n}}{b_{n}}}\leq \limsup _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}.$ 