Cfrgtkky

1 $begingroup$ Suppose that the domain of convergence of the power series $sum_{k=0}^{infty} c_{k}x^{k}$ contains the interval $(-r, r)$ . Define $$f(x) = sum_{k=0}^{infty} c_{k}x^{k} hspace{1cm} text{ > if } |x| < r. $$ Let $[a, b] subseteq (-r, r).$ Prove that $$int_{a}^{b} f(x) mathop{dx} = sum_{k = 0}^{infty} frac{c_{k}}{k + 1}left(b^{k + 1} - a^{k + 1}right).$$ Here's the solution I have. It might be wrong because it's not official. Recall Theorem $5$ , which states that if a sequence of integrable functions ${f_{n} : [a, b] rightarrow mathbb{R}}$ converges uniformly to the function $f : [a, b] rightarrow mathbb{R}$ , then the limit function is also integrable. So, $$int_{a}^{b} f(x) mathop{dx} = lim_{ntoinfty} int_{a}^{b} sum_{k = 0}^{n} c_{k}x^{k} = lim_{nt...