# Thread: Uniform Convergence of Power Series

1. ## Uniform Convergence of Power Series

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

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

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

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

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

I am unable to show that $\displaystyle {a_n}r^n$ 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. $\displaystyle |a_n|r^n$ is NOT $\displaystyle |a_nr^n|$.