manipulating convergent series ManipulatingConvergentSeries 2004-11-23 15:37:10 2006-10-14 15:10:36 Theorem convergent series % 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 \theoremstyle{definition} \newtheorem{thmplain}{Theorem} The \PMlinkescapetext{terms} of the series in the following theorems are supposed to be either real or complex numbers. \begin{thmplain} \, If the series\, $a_1+a_2+\cdots$\, and\, $b_1+b_2+\cdots$\, converge and have the sums $a$ and $b$, respectively, then also the series \begin{align} (a_1+b_1)+(a_2+b_2)+\cdots \end{align} converges and has the sum $a\!+\!b$. \end{thmplain} {\em Proof.} \,The $n^\mathrm{th}$ partial sum of (1) has the limit $$\lim_{n\to\infty}\sum_{j = 1}^n(a_j+b_j) = \lim_{n\to\infty}\sum_{j = 1}^na_j+\lim_{n\to\infty}\sum_{j = 1}^nb_j = a\!+\!b.$$ \begin{thmplain} \, If the series\, $a_1+a_2+\cdots$\, converges having the sum $a$ and if $c$ is any \PMlinkescapetext{constant}, then also the series \begin{align} ca_1+ca_2+\cdots \end{align} converges and has the sum $ca$. \end{thmplain} {\em Proof.} \,The $n^\mathrm{th}$ partial sum of (2) has the limit $$\lim_{n\to\infty}\sum_{j = 1}^nca_j = c\lim_{n\to\infty}\sum_{j = 1}^na_j = ca.$$ \begin{thmplain} \, If the \PMlinkescapetext{terms} of any converging series \begin{align} a_1+a_2+a_3+\cdots \end{align} are grouped arbitrarily without changing their \PMlinkescapetext{order}, then the resulting series \begin{align} (a_1+\cdots+a_{m_1})+(a_{m_1+1}+\cdots+a_{m_2})+(a_{m_2+1}+\cdots+a_{m_3})+\cdots \end{align} also converges and its sum equals to the sum of (3). \end{thmplain} {\em Proof.} \,Since all the partial sums of (4) are simultaneously partial sums of (3), they have as limit the sum of the series (3).