Let's consider the function $$ f\left( x \right) = \sum\limits_{k = 1}^\infty {\frac{1} {{n^2 + x^2 n^3 }}} $$ I proved that the partial sums $S_N=\sum_{k=1}^N f_k$ converges uniformly on $\Bbb R$ where $ f_n(x)= \frac{1}{n^2+x^2n^3}$ . I also proved that the sequence $S_N'=\sum_{k=1}^N f'_k$ converges pointwise on $\Bbb R$ and also uniformly on sets of the form $ \left| x \right| \geqslant \delta > 0 $.
I want to prove that $f$ is differentiable in sets of the form $ \left| x \right| \geqslant \delta > 0 $ and that the series of the derivates i.e $S_N'$ converges to the function $f'$
And finally but less important prove that $f'(0)$ exist and equals zero.
I don't know how to proceed here, because I'm working on non compact sets :S , so in genral it's more difficult
