# Series - Power series

Real number : x , $x_0$

Power series : $\displaystyle\sum_{n=0}^\infty a_nx_n \,, \displaystyle\sum_{n=0}^\infty a_n(x-x_0)^n$

Whole number : n

$\displaystyle\sum_{n=0}^\infty a_n x^n=a_0+a_1x+a_2x^2+...+a_nx^n+...$
2. Power Series in $(x-x_0)$
$\displaystyle\sum_{n=0}^\infty a_n (x-x_0)^n=a_0+a_1 (x-x_0)+a_2 (x-x_0)^2+...+a_n (x-x_0)^n+...$
$f(x)= \displaystyle\sum _{n=0}^\infty a_n(x-x_0)^n$ is convergent is called the interval of convergence.
If the interval of convergence is $(x_0-R, x_0+R)$ for  some $R\ge 0,$ the R is called the radius of convergence. It is given as
$R= \displaystyle\lim_{n \to \infty} {1 \over \sqrt [n]{a_n}}$  or  $R= \displaystyle\lim_{n \to \infty} \bigg | { a_n \over a_{n+1} } \bigg | .$