# 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 $l$.

Remark. This theorem is also valid if $a_{n}$ and $b_{n}$ are monotone, tending to $0$.

Title Stolz-Cesaro theorem StolzCesaroTheorem 2013-03-22 13:17:16 2013-03-22 13:17:16 CWoo (3771) CWoo (3771) 9 CWoo (3771) Theorem msc 40A05 CesaroMean ExampleUsingStolzCesaroTheorem KroneckersLemma