Stolz-Cesaro theorem

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

$$\underset{n\to \mathrm{\infty }}{lim}\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=l$$

Then the limit:

$$\underset{n\to \mathrm{\infty }}{lim}\frac{a_n}{b_n}$$

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
Canonical name	StolzCesaroTheorem
Date of creation	2013-03-22 13:17:16
Last modified on	2013-03-22 13:17:16
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	9
Author	CWoo (3771)
Entry type	Theorem
Classification	msc 40A05
Related topic	CesaroMean
Related topic	ExampleUsingStolzCesaroTheorem
Related topic	KroneckersLemma