I'm posting this again because the other was plagued with errors in the first post. My fault and I apologize. I'll do a better job this time, I hope.(adsbygoogle = window.adsbygoogle || []).push({});

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

Let f(n,x) = [tex]

\sum\limits_{n = 1}^{\inf } {( - 1)^n (1-x^{2})x^{n}}

[/tex]

a) Test for absolutely convergence on [0,1]

b) Test for uniformly convergence on [0,1]

c) Is [tex]

\sum\limits_{n = 1}^{\inf } {|f(n,x)|}

[/tex] uniformly convergent on [0,1]?

2. Relevant equations

N/A

3. The attempt at a solution

Alright, a) is simple actually, a simple application of the root/ratio test will do, but here comes the catch.

I know that b) is true, since a power series is uniformly convergent inside its radius of convergence and the endpoint in this case is trivial. It just seems, however, that with a comparison with a geometric series, I can use the Weierstrass M-Test to prove all 3 at once.

For any 'x' in [0,1), I can choose an a such that 0 <= x <= a <1 by the density of R. Therefore

[tex]

|\sum\limits_{n = 1}^{\inf } {f(n,x)} | <= \sum\limits_{n = 1}^{\inf } {|f(n,x)|} < \sum\limits_{n = 1}^{\inf } {(a)^n} = \frac{a}{1-a}

[/tex]

and for x = 1 and any a > 0

[tex]

|\sum\limits_{n = 1}^{\inf } {|f(n,x)|} | <= \sum\limits_{n = 1}^{\inf } {|f(n,x)|} = 0 < \sum\limits_{n = 1}^{\inf } {(a)^n} = \frac{a}{1-a}[/tex]

I have just chosen the function to be uniformally cauchy on [0,1) U {1} (have I?) by a basic comparison test with a series which is uniformly cauchy on (0,1) and by considering the end point separately.

But it also seems that the Weierstrass M-Test gives me "Absolutely Uniformly Convergence" (that is, the absolute value of the function is also uniformly convergent) for free. Am I right? Is that always gonna hold true, or is this a property of power series?

**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: Weierstrass M-Test and Absolutely Uniformly Convergence

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