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 (the $∙/\infty$ 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}}}\

also exists and it is equal to ℓ.

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:^[1] 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}}}.

References

Mureşan, Marian (2008), A Concrete Approach to Classical Analysis, Berlin: Springer, pp. 85–88, ISBN 978-0-387-78932-3 .
Stolz, Otto (1885), Vorlesungen über allgemeine Arithmetik: nach den Neueren Ansichten, Leipzig: Teubners, pp. 173–175 .
Cesàro, Ernesto (1888), "Sur la convergence des séries", Nouvelles annales de mathématiques, Series 3, 7: 49–59 .
Pólya, George; Szegő, Gábor (1925), Aufgaben und Lehrsätze aus der Analysis, I, Berlin: Springer .
A. D. R. Choudary, Constantin Niculescu: Real Analysis on Intervals. Springer, 2014, ISBN 9788132221487, pp. 59-62
J. Marshall Ash, Allan Berele, Stefan Catoiu: Plausible and Genuine Extensions of L’Hospital's Rule. Mathematics Magazine, Vol. 85, No. 1 (February 2012), pp. 52–60 (JSTOR)

External links

Notes

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

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