$$ f(x) = \sum \limits_{n=1}^\infty \frac{\arctan(n + x)}{n^2+x^2} $$
For which x function f is continuous and for which is differentiable? if $f'(0) $ exists than does $f'(0)>0$?
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1$|f(x)|\leq\frac{\pi}{2}\sum\frac{1}{n^2}<\infty$ so the series converges uniformly on $\mathbb{R}$. – yoyo Sep 05 '13 at 03:48
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A related problem. – Mhenni Benghorbal Sep 05 '13 at 04:48
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Since $|\arctan y|<{\pi\over2}$ for all real $y$ it follows that the termwise differentiated series $$\Sigma'(x):=\sum_{n=1}^\infty{-2x\arctan(n+x)\over(n^2+x^2)^2}+{1\over\bigl(1+(n+x)^2\bigr)(n^2+x^2)}$$ is uniformly convergent on any compact interval. Therefore $f$ is everywhere differentiable, and $f'(x)$ is given by $\Sigma'(x)$. In particular it follows that $f'(0)>0$.
Christian Blatter
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