Exercise 6.6.3

We are given $f(x)={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{n=0}^{\infty }{a}_n{x}^n$ defined on some nontrivial interval $(-R,R)$. $a_0=f(0)$ by definition, now applying Theorem 6.5.7 we get

and $f^{\prime }(0)=a_1\cdot 1$ by definition of $f^{\prime }$, continue like this applying 6.5.7 each time to get