Stolz–Cesàro theorem

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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 (the ∙/∞ case)[edit]

Let and be two sequences of real numbers. Assume that is strictly monotone and divergent sequence (i.e. strictly increasing and approaches or strictly decreasing and approaches ) and the following limit exists:

Then, the limit

also exists and it is equal to .

History[edit]

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[edit]

The general form of the Stolz–Cesàro theorem is the following:[1] If and are two sequences such that is monotone and unbounded, then:

References[edit]

External links[edit]

Notes[edit]

  1. ^ l'Hôpital's rule and Stolz-Cesàro theorem at imomath.com

This article incorporates material from Stolz-Cesaro theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.