Stolz–Cesàro theorem

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 unbounded 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 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 PS.

References

• Marian Mureşan: A Concrete Approach to Classical Analysis. Springer 2008, ISBN 9780387789323, p. 85 (restricted online copy, p. 85, at Google Books)

S : Stolz, O. Vorlesungen über allgemeine Arithmetik: nach den Neueren Ansichten, Teubners, Leipzig, 1885, pp. 173--175.
C : Cesaro, E. , Sur la convergence des séries, Nouvelles annales de mathématiques Series 3, 7 (1888), 49--59.
PS : Pólya, G. and Szegö, G. Aufgaben und Lehrsätze aus der Analysis, v. 1, Berlin, J. Springer, 1925.

External links

• Proof of Stolz–Cesàro theorem

Stolz-Cesaro theorem at PlanetMath.