$f\left(x\right)=\underset{n=0}{\overset{\infty }{\sum }}{c}_{n}{\left(x-a\right)}^{n}$
the radius of convergence $r$ is a nonnegative real number (or possibly, infinity), such that the series converges if
$|\text{\hspace{0.17em}}x-a\text{\hspace{0.17em}}|
$|\text{\hspace{0.17em}}x-a\text{\hspace{0.17em}}|>r$ .