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

Radious of convergence : R

  1. Power Series in x 
    \(\displaystyle\sum_{n=0}^\infty a_n x^n=a_0+a_1x+a_2x^2+...+a_nx^n+...\)
  2. Power Series in \((x-x_0)\)
    \(\displaystyle\sum_{n=0}^\infty a_n (x-x_0)^n=a_0+a_1 (x-x_0)+a_2 (x-x_0)^2+...+a_n (x-x_0)^n+...\)
  3. Interval of convergence 
    The set of those values of x for which the function 
    \(f(x)= \displaystyle\sum _{n=0}^\infty a_n(x-x_0)^n\) is convergent is called the interval of convergence. 
  4. Radius 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  \(R= \displaystyle\lim_{n \to \infty} \bigg | { a_n \over a_{n+1} } \bigg | .\)