View source for Stolz–Cesàro theorem

In [[mathematics]], the '''Stolz–Cesàro theorem''' is a criterion for proving the [[convergent sequence|convergence of a sequence]].  The theorem is named after [[mathematician]]s [[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 {{math|∙/∞}} case ==
Let  <math>(a_n)_{n \geq 1}</math> and  <math>(b_n)_{n \geq 1}</math> be two [[sequence]]s of [[real number]]s. Assume that <math>(b_n)_{n \geq 1}</math> is a [[Monotonic function|strictly monotone]] and divergent sequence (i.e. [[monotonically increasing|strictly increasing]] and approaching <math> + \infty </math>, or [[monotonically decreasing|strictly decreasing]] and approaching <math> - \infty </math>) and the following [[limit of a sequence|limit]] exists:
:<math> \lim_{n \to \infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}=l.\ </math>
Then, the limit
:<math> \lim_{n \to \infty} \frac{a_n}{b_n}=l.\ </math>

== Statement of the theorem for the {{math|0/0}} case ==
Let  <math>(a_n)_{n \geq 1}</math> and  <math>(b_n)_{n \geq 1}</math> be two [[sequence]]s of [[real number]]s. Assume now that <math>(a_n)\to 0</math> and  <math>(b_n)\to 0</math> while <math>(b_n)_{n \geq 1}</math> is  [[Monotonic function|strictly monotone]]. If 
:<math> \lim_{n \to \infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}=l,\ </math>
then
:<math> \lim_{n \to \infty} \frac{a_n}{b_n}=l.\ </math><ref>https://www.springer.com/gp/book/9788132221470 see pages 59-60</ref>

== History ==
The ∞/∞ case is stated and proved on pages 173—175 of Stolz's 1885 book and also on page 54 of Cesàro's 1888 article. 

It appears as Problem 70 in Pólya and Szegő (1925).

== The general form ==
The general form of the Stolz–Cesàro theorem is the following:<ref>[http://www.imomath.com/index.php?options=686 l'Hôpital's rule and Stolz-Cesàro theorem at imomath.com]</ref> If <math> (a_n)_{n\geq 1}</math> and <math> (b_n)_{n\geq 1}</math> are two sequences such that <math>(b_n)_{n \geq 1}</math> is monotone and unbounded, then:
:<math>\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}.</math>

==References==
*{{citation
 | last = Mureşan | first = Marian
 | isbn = 978-0-387-78932-3
 | location = Berlin
 | pages = 85–88
 | publisher = Springer
 | title = A Concrete Approach to Classical Analysis
 | url = https://books.google.com/books?id=5iK9OX9z014C&pg=PA85
 | year = 2008}}.
*{{citation
 | last = Stolz | first = Otto | author-link = Otto Stolz
 | location = Leipzig
 | pages = 173–175
 | publisher = Teubners
 | title = Vorlesungen über allgemeine Arithmetik: nach den Neueren Ansichten
 | url = https://archive.org/stream/vorlesungenbera01stolgoog#page/n181/mode/1up
 | year = 1885}}.
*{{citation
 | last = Cesàro | first = Ernesto | author-link = Ernesto Cesàro
 | journal = [[Nouvelles Annales de Mathématiques|Nouvelles annales de mathématiques]]
 | pages = 49–59
 | series = Series 3
 | title = Sur la convergence des séries
 | volume = 7
 | year = 1888}}.
*{{citation
 | last1 = Pólya | first1 = George | author1-link = George Pólya
 | last2 = Szegő | first2 = Gábor | author2-link = Gábor Szegő
 | location = Berlin
 | publisher = Springer
 | title = Aufgaben und Lehrsätze aus der Analysis
 | volume = I
 | year = 1925}}.
*A. D. R. Choudary, Constantin Niculescu: ''Real Analysis on Intervals''. Springer, 2014, {{isbn|9788132221487}}, pp. [https://books.google.com/books?id=I8aPBQAAQBAJ&pg=PA59 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.&nbsp;52–60 ([https://www.jstor.org/stable/10.4169/math.mag.85.1.52 JSTOR])

== External links ==
* [http://www.imomath.com/index.php?options=686 l'Hôpital's rule and Stolz-Cesàro theorem at imomath.com]
*{{PlanetMath|ProofOfStolzCesaroTheorem|Proof of Stolz–Cesàro theorem}}

==Notes==
<references/>

{{PlanetMath attribution|id=3774|title=Stolz-Cesaro theorem}}

{{DEFAULTSORT:Stolz-Cesaro theorem}}
[[Category:Theorems in real analysis]]
[[Category:Convergence tests]]