# Continuity of a power-series function

1. Jan 25, 2013

### chi8

1. The problem statement, all variables and given/known data

Prove the function:
$g(x)=\sum_{n=1}^{\infty }\frac{1}{^{n^{0.5}}}(x^{2n}-x^{2n+1})$

is continuous in [0,1].

2. The attempt at a solution

I tried to look at this functions as:

$g(x)=(1-x)\sum_{n=1}^{\infty }\frac{1}{^{n^{0.5}}}x^{2n}$

but I couldn't find a way solving it from here.
Finding the radius of convergence (which is 1) didn't help a lot...

2. Jan 25, 2013

### jbunniii

Yes, writing it as
$$g(x) = (1-x) \sum_{n=1}^\infty \frac{1}{n^{0.5}} x^{2n}$$
is a good start. Let us introduce the notation
$$p(x) = \sum_{n=1}^\infty \frac{1}{n^{0.5}} x^{2n}$$
This is a power series. If its radius of convergence is 1, do you know a theorem which tells you that p is continuous on (-1,1)? If so, then you only need to worry about the point x = 1. Does the series converge at x = 1?