showing convergence of $\int f(x)dx$ if $f(x)$ converges

Question

Given $f(x) = \sum_{n=1}^\infty a_n x^n$ converges in $(-R, R)$. How do I show that $$ \sum_{n=1}^\infty \frac {a_n}{n+1} x^{n+1} $$ converge in $(-R, R)$?

My idea is to show $$ \left | \sum_{k=m+1}^n \frac{a_k}{k+1}x^{k+1} \right | \le R \left |\sum_{k=m+1}^n a_k x^k \right | $$ But I am not sure if that inequality holds for $x\in (-R, R)$.

Answer 1

$ \lim \sup |\frac{a_n}{n+1}|^{1/n}= \lim \sup |a_n|^{1/n}$, hence $\sum_{n=1}^\infty a_n x^n$ and $\sum_{n=1}^\infty \frac {a_n}{n+1} x^{n+1}$ have the same radius of convergence.

Answer 2

The assertion “the series $\sum_{n=1}^\infty a_nx^n$ converges in $(-R,R)$” is equivalent to $\limsup_n\sqrt[n]{\lvert a_n\rvert}\geqslant\frac1R$. But$$\limsup_n\sqrt[n]{\lvert a_n\rvert}=\limsup_n\sqrt[n]{\left\lvert\frac{a_{n+1}}{n+1}\right\rvert}.$$