Radius of convergence of derivative of power series equals radius of convergence

Statement

Consider a power series:

$\sum _{k=0}^{\infty }a_{k}(x-\mu )^{k}$

The radius of convergence of this power series equals the radius of convergence of its derivative, which is:

$\sum _{k=0}^{\infty }(k+1)a_{k+1}(x-\mu )^{k}$