sum of series SumOfSeries 2004-11-18 16:44:50 2007-12-02 16:17:34 Definition series partial sum % 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 If a series $\sum_{n = 1}^\infty a_n$ of real or complex numbers is convergent and the limit of its partial sums is $S$, then $S$ is the {\em sum of the series}.\, This circumstance may be denoted by $$\sum_{n = 1}^\infty a_n = S$$ or equivalently $$a_1+a_2+a_3+\cdots = S.$$ Nevertheless, one should not think that this means an addition of infinitely many numbers --- it's only a question of the limit $$\lim_{n\to\infty}\underbrace{(a_1+a_2+\cdots+a_n)}_{\textrm{partial sum}}.$$ The sum of the series is equal to the sum of a partial sum and the corresponding remainder term.