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 Rudin 【Principle of Mathematical Analysis】Notes & Solutions
01-27-2015, 12:04 PM (This post was last modified: 12-14-2020 06:53 PM by elim.)
Post: #331
 elim Moderator Posts: 582 Joined: Feb 2010 Reputation: 0
Chapter VIII SOME SPECIAL FUNCTIONS.
POWER SERIES

$\quad$In this section we shall derive some properties of functions which are represented by power seires
$\displaystyle{ f(x) = \sum_{n=0}^{\infty} c_n x^n \tag{1}}$
$\quad$or, more generally,
$\displaystyle{ f(x) = \sum_{n=0}^{\infty}c_n (x -a)^n. \tag{2}}$
$\quad$There are called analytic functions.
$\quad$We shall restrict ourselves to real values of $x$ hence we shall encounter intervals of convergence
$\quad$instead of circles of convergence (see Th.3.39)
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