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 .

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.