Let $f(x)=\sum\limits_{n=1}^{\infty}\frac{1}{1+n^2|x|}$.
- For what values of x does this series converge.
- Characterize the subsets of $\mathbb{R}$ on which the series converges uniformly.
There are only two cases to deal with $x=0$ and $x>0$. If $x<0$, because of the absolute value, we just have the $x>0$ case again. For $x>0$ we have $$\sum\limits_{n=1}^{\infty}\frac{1}{1+n^2|x|}\leq \sum\limits_{n=1}^{\infty}\frac{1}{n^2|x|}=\frac{1}{|x|}\sum\limits_{n=1}^{\infty}\frac{1}{n^2}<\infty.$$ For $x=0$ we have a divergent series.
On any interval that contains $0$ convergence will not occur. On any inverval that does not contain $0$ there is uniform convergence by the Weierstrass M-Test.
If we have an open interval where $0$ is an endpoint, then we don't have uniform convergence. See,
$$\left|\sum\limits_{n=1}^{\infty}\frac{1}{1+n^2|x|}\right|<\frac{1}{2}$$
and let $x=\pm \frac{1}{n^2}$.
Are my above answers correct?