Differentiation of Power Series

Definition

Suppose a power series $f(x)=\sum _{n=0}^{\infty }{a}_n(x-c{)}^n$ has a Radius of Convergence $R$ Then, $f$ is differentiable on a $(c-R,c+R)$, and the derivative of $f$ is $f^{\prime }(x)=\sum _{n=1}^{\infty }n{a}_n(x-c{)}^{n-1}$