Weierstrass M-test

[Nemesis10192]

So I want to show that f(x)=sum from n=1 to infinity [1/(n(1+nx^2))] is uniformly convergent on the interval (a,b) where a>0 and b>a, or b<0 and a<b but not on the interval (0,b).

I want to use the Weierstrass M-test to show this, which states that assuming f_n(x)<=M_n for all natural n and x in [a,b] and that the sum from n=1 to infinity of M_n is convergent, then sum from n=1 to infinity of f_n(x) is uniformly convergent on [a,b].

So I have f_n(x)=1/(n(1+nx^2)) < 1/(n^2x^2) but I can't say this is < 1/n^2 since that is only true if |x|>1 which will not always be true...

Basically M_n needs to depend only on n so I'm unsure how to bound the function...any help?!

 

[Ishuda]

So I want to show that f(x)=sum from n=1 to infinity [1/(n(1+nx^2))] is uniformly convergent on the interval (a,b) where a>0 and b>a, or b<0 and a<b but not on the interval (0,b).

I want to use the Weierstrass M-test to show this, which states that assuming f_n(x)<=M_n for all natural n and x in [a,b] and that the sum from n=1 to infinity of M_n is convergent, then sum from n=1 to infinity of f_n(x) is uniformly convergent on [a,b].

So I have f_n(x)=1/(n(1+nx^2)) < 1/(n^2x^2) but I can't say this is < 1/n^2 since that is only true if |x|>1 which will not always be true...

Basically M_n needs to depend only on n so I'm unsure how to bound the function...any help?!

First, we see that that the function is an even function [ f(x) = f(-x) ] so that, if f converges in a region (a,b); b > a > 0, then f converges in (-b, -a). So we only need to consider b > a > 0 and the regions (a, b) and (0, b).

For convergence: If a > 0, b > a and x belongs to (a, b) then
Since |a|\(\displaystyle \ne\) 0, a² > 0 and, if x > a then x² > a² so that, if n > 0, n² x² > n² a² and \(\displaystyle \frac{1}{n^2 a^2} \gt \frac{1}{n^2 x^2}\), etc. ending with something like M_n = \(\displaystyle \frac{1}{n^2 a^2}\)

For non-uniform convergence what happens if a is not bounded away from zero?