1. ## Uniform COnvergence

For $\displaystyle f(x)=\sum_{n=1}^{\infty} \frac{1}{n(1+nx^2}$, on what intervals of x in the form $\displaystyle (a,b)$ does the series converge uniformly?

$\displaystyle \left| \frac{1}{n(1+nx^2)} \right| \leq \left| \frac{1+nx^2}{n(1+nx^2)} \right|= \left| \frac{1}{n} \right|$

I would now choose my $\displaystyle M_n=\frac{1}{n}$. However, $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}$ does not converge!

I can't see another way of doing it so that I get it to be less than a convergent sequence. =S

2. Originally Posted by Showcase_22
For $\displaystyle f(x)=\sum_{n=1}^{\infty} \frac{1}{n(1+nx^2}$, on what intervals of x in the form $\displaystyle (a,b)$ does the series converge uniformly?

$\displaystyle \left| \frac{1}{n(1+nx^2)} \right| \leq \left| \frac{1+nx^2}{n(1+nx^2)} \right|= \left| \frac{1}{n} \right|$

I would now choose my $\displaystyle M_n=\frac{1}{n}$. However, $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}$ does not converge!

I can't see another way of doing it so that I get it to be less than a convergent sequence. =S

$\displaystyle \left |\frac{1}{n(1+nx^2)} \right|=\frac{1}{n+n^2x^2}\leq \frac{1}{n^2x^2}$

As the series $\displaystyle \frac{1}{x^2}\sum\limits_{n=1}^\infty\frac{1}{n^2}$ converges no matter what is $\displaystyle x \neq 0$ you're done.

Tonio