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 and be two sequences of real numbers. Assume that is a strictly monotone and divergent sequence and the following limit exists:
Then, the limit

Statement of the theorem for the case

Let and be two sequences of real numbers. Assume now that and while is strictly monotone. If
then

History

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

The general form

The general form of the Stolz–Cesàro theorem is the following: If and are two sequences such that is monotone and unbounded, then: