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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

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new correction: Error in proof of Proposition 2 by alex2907

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new question: Banach lattice valued Bochner integrals by math ias

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new question: young tableau and young projectors by zmth

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new question: binomial coefficients: is this a known relation? by pfb

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new question: difference of a function and a finite sum by pfb