I'd like to determine whether $F(x)=\sum^{\infty}_{n=1}\frac{x}{n(1+nx^2)}$ converges for all $x\in\Bbb R$
Using the M-test:
Finding $||f_n(x)||_{\infty}$:
$f'_n(x)=\frac{x'n(1+nx^2)-2n^2x^2}{(n(1+nx^2))^2}=\frac{n-n^2x^2}{(n+n^2x^2)^2}=0$
So, $x=+-\frac{1}{\sqrt{n}}$
Hence, $|f_n(\sqrt{n}^{-1})|=\frac{1/\sqrt{n}}{n(1+n/n)}=\frac{1}{2n^{3/2}}$ and hence $\sum^{\infty}_0||f_n(x)||_{\infty}<\infty$
So the sum converges uniformly. If it converge uniformly, then by this theorem it's continuous?:
