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 for 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

Statement of the theorem for the 0/0 case[edit]

Let and be two sequences of real numbers. Assume now that and while is strictly monotone. If




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:[2] If and are two sequences such that is monotone and unbounded, then:


  • 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[edit]


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