# Math Help - Uniform Convergence Proof

1. ## Uniform Convergence Proof

I'm having trouble getting this proof, uniform convergence has always been a hard thing for me.

If Σk=0infinity ak converges absolutely, prove that Σk=0infinity akxk converges uniformly on [-1,1]

Since this is a power series, I thought I'd go the approach that if akxk converges absolutely for x=1, then it converges uniformly for [-1,1]. I'm not sure this is the right approach though, since I'm having trouble starting.

2. ## Re: Uniform Convergence Proof

It's simple: the remainder $\left|\sum_{k=N}^\infty a_kx^k\right| \le \sum_{k=N}^\infty \left|a_kx^k\right| \le \sum_{k=N}^\infty \left|a_k\right| < \varepsilon$ for a sufficiently large N.