Chapter 9.10, Problem 46ES

Prove: if the power series ${\displaystyle {\sum }_{k=0}^{\infty }{a}_k{x}^k}$ and ${\displaystyle {\sum }_{k=0}^{\infty }{b}_k{x}^k}$ have the same sum on an interval $(-r,r),$ then $a_k=b_k$ for all values of $k.$