Sequences, series, summability Sequences and limits Theorem

Stolz–Cesàro theorem

Builds on Limit of a sequence Monotone sequence Sequence tending to infinity

Statement

Let $(a_n)$ and $(b_n)$ be real sequences and let $L$ be a real number. Assume that $(b_n)$ is strictly increasing and that $b_n \to +\infty$ as $n \to \infty$ . Assume that $\lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L.$ Then $a_n / b_n \to L$ as $n \to \infty$ .

Remarks

There is also a $0/0$ form. If $a_n \to 0$ , $b_n \to 0$ , $(b_n)$ is strictly decreasing and the quotient of the differences tends to $L$ , then $a_n / b_n \to L$ . With $b_n = n$ it gives the convergence of the Cesàro means. For example, with $a_n = 1 + 2 + \cdots + n$ and $b_n = n^2$ , the quotient of differences is $(n+1)/(2n+1) \to 1/2$ , so $a_n / n^2 \to 1/2$ . It is a discrete analogue of L'Hôpital's rule.

History

Otto Stolz proved the $\infty/\infty$ case in a book of 1885. Ernesto Cesàro gave it in an article of 1888.

Reading path

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Sequence definition
Real sequence definition
Limit of a sequence definition
Monotone sequence definition
Sequence tending to infinity definition
Stolz–Cesàro theorem statement

Tag	`0Q3AKZV74D`
URI	`m2://0Q3AKZV74D`
Kind	statement, theorem
Writing	W1 stated
MSC	40A05
Status	This copy of the site has no build data. Run `m2 status 0Q3AKZV74D`.
Uses	Limit of a sequence, Monotone sequence, Sequence tending to infinity
Same as	Wikidata Q1052752 Wikipedia Stolz–Cesàro theorem

Stolz–Cesàro theorem

Statement

Remarks

History

Reading path

Further reading

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