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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

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This is one of the properties about uniform convergence. See for example, p. 8 of math.uga.edu/~pete/243functions1.pdf –  random Nov 22 '12 at 23:09
    
I can addapt the argument in the proof? because he assume that the convergence is on a compact interval $[a,b]$ –  Joseph Nov 23 '12 at 0:18
    
I think that I did it. –  Joseph Nov 23 '12 at 3:07
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