# Thread: Uniform Convergence of Power Series

1. ## Uniform Convergence of Power Series

Suppose $\sum_{n=0}^{\infty} {a_n}x^n$ converges on $[-R, R]$ Suppose $\delta < R.$ Show that $\sum_{n=0}^{\infty} {a_n}x^n$ converges uniformly on $[-\delta, \delta]$.

Hint given: Pick $r$ such that $\delta < r < R.$ You are given that $\sum_{n=0}^{\infty} {a_n}r^n$ converges, therefore $|a_n|r^n \rightarrow 0$. In particular, $|a_n|r^n \leq C$, where $C$ is independent of $n$.

I am unable to show that ${a_n}r^n$ converges absolutely, while the hint suggests that I should show this. Thanks in advance for any help!

2. Originally Posted by h2osprey
Suppose $\sum_{n=0}^{\infty} {a_n}x^n$ converges on $[-R, R]$ Suppose $\delta < R.$ Show that $\sum_{n=0}^{\infty} {a_n}x^n$ converges uniformly on $[-\delta, \delta]$.

Hint given: Pick $r$ such that $\delta < r < R.$ You are given that $\sum_{n=0}^{\infty} {a_n}r^n$ converges, therefore $|a_n|r^n \rightarrow 0$. In particular, $|a_n|r^n \leq C$, where $C$ is independent of $n$.

I am unable to show that ${a_n}r^n$ converges absolutely, while the hint suggests that I should show this. Thanks in advance for any help!
I can see nothing here that suggests absolute convergence. $|a_n|r^n$ is NOT $|a_nr^n|$.