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**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?

My answer:

$\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