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# Stolz-Cesaro theorem

Let $(a_{n})_{{n\geq 1}}$ and $(b_{n})_{{n\geq 1}}$ be two sequences of real numbers. If $b_{n}$ is positive, strictly increasing and unbounded and the following limit exists:

$\lim_{{n\rightarrow\infty}}\frac{a_{{n+1}}-a_{n}}{b_{{n+1}}-b_{n}}=l$ |

Then the limit:

$\lim_{{n\rightarrow\infty}}\frac{a_{n}}{b_{n}}$ |

also exists and it is equal to $l$.

Keywords:

convergence,sequence,limit

Related:

CesaroMean, ExampleUsingStolzCesaroTheorem, KroneckersLemma

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

40A05*no label found*

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## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

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new question: Lorenz system by David Bankom

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new correction: examples and OEIS sequences by fizzie

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new correction: Define Galois correspondence by porton

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new correction: Closure properties on languages: DCFL not closed under reversal by babou

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new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag