24 - Differentiation Theorem for Real Power Series

Differentiation Theorem for real power series

Let $\sum c_k{x}^k$ be a real power series with Radius of Convergence $R$
Assume that $0<R\le \infty$
For $|x|<R$

$$f(x)=\sum _{k=0}^{\infty }{c}_k{x}^k$$

Then $f(x)$ is well defined whenever $|x|<R$. And the derivative $f^{\prime }(x)$ exists

$$f^{\prime }(x)=\frac{d}{dx}\left(\sum _{k=0}^{\infty }{c}_k{x}^k\right)=\sum _{k=0}^{\infty }\frac{d}{dx}(c_k{x}^k)=\sum _{k=1}^{\infty }k{c}_k{x}^{k-1}$$