Stolz-Cesaro theorem StolzCesaroTheorem 2002-12-18 16:53:43 2007-08-22 20:24:44 Theorem convergence sequence limit % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 $l$. \textbf{Remark.} This theorem is also valid if $a_n$ and $b_n$ are monotone, tending to $0$.