I have to check for Uniform Convergence of $\sum_{n=1}^{\infty} \frac{nx}{1+n^2x^2}$.
Here's what I have done so far.
Take $\epsilon = 1/2$ , $x = 0$.
$S_{1} = 0 \space \forall \space n \geq 1$
Hence $S_{1} < \epsilon \space \forall \space n \geq 1$
Hence for the series to be uniformly convergent, for any value of x, $S_{n} \leq \epsilon$ $\forall \space n \geq 1$
Take $x = 1$. $S_{1} = n/(1+n^2) = 1/2 $ when $n = 1$, which is clearly not $\leq \epsilon$.
Hence the series is not uniformly convergent.
Is there something wrong in my approach? I feel there could be a mistake because $S_1 \leq \epsilon$ for $n\geq 1$ does not mean that we have found an m for which $S_n \leq \epsilon \space \forall n \geq m.$ What should the correct approach be?
this series does not converge for $x\neq 0$. we can use the limit comparison test for $\sum_{n=1}^{\infty} \frac{nx}{1+n^2x^2} $ and $\sum_{n=1}^{\infty} \frac{1}{n}$. $\lim_{n\to\infty} \frac{\frac{nx}{1+n^2x^2}}{\frac{1}{n}} = \lim_{n\to\infty} \frac{n^{2}x}{1+n^2x^2} = \lim_{n\to\infty} \frac{x}{\frac{1}{n^2}+x^2} = \frac{1}{x} \neq 0 $. and thus from the limit comparison test the series $\sum_{n=1}^{\infty} \frac{nx}{1+n^2x^2} $ and $\sum_{n=1}^{\infty} \frac{1}{n}$ converge and diverge together, we know that the harmonic series - $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges and thus we can deduce that the series $\sum_{n=1}^{\infty} \frac{nx}{1+n^2x^2} $ diverges for all $x \neq 0$