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 $b_{n}$ is strictly increasing and approaches infinity 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 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 $b_{n}$ 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.

References

Marian Mureşan: A Concrete Approach to Classical Analysis. Springer 2008, ISBN 978-0-387-78932-3, p. 85 (restricted online copy, p. 85, at Google Books)
Stolz, O. Vorlesungen über allgemeine Arithmetik: nach den Neueren Ansichten, Teubners, Leipzig, 1885, pp. 173–175. (online copy at Internet Archive)
Cesaro, E., Sur la convergence des séries, Nouvelles annales de mathématiques Series 3, 7 (1888), 49—59.
Pólya, G. and Szegö, G. Aufgaben und Lehrsätze aus der Analysis, v. 1, Berlin, J. Springer 1925.

External links

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

Proof of Stolz–Cesàro theorem at PlanetMath

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