(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove:

(1) the series

[tex]\sum_{n=0}^\infty (-1)^n x^n (1-x)[/tex]

converges absolutely and uniformly on the interval [0,1]

(2) the series

[tex]\sum_{n=0}^\infty x^n (1-x)[/tex]

converges absolutely and uniformly on the interval [0,1]

3. The attempt at a solution

I have shown, by induction, that the limiting function of the second series is 1 - x^{n+1}, which goes to 1. Thus the series of functions converges (absolutely, since all values are positive) but is 0 at x = 1, so thus not continuous. Therefore, the convergence of the second series is not uniform. However, this also shows that the first series converges absolutely.

Where I am stuck is with uniform convergence of the first series. Using partial sums I was able to show that the series converges to (1-x)/(1+x), but how do I show this is uniform? I don't think it's enough to say that the limiting function is continuous in the given interval.

Can anyone tell me if I'm on the right track or what I can use to prove uniform convergence? Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Uniform convergence of a series of functions

**Physics Forums | Science Articles, Homework Help, Discussion**