Why does arctan(1/n) summed not converge? My question is as follows: WolframAlpha tells me that the sum series S doesn't converge, but why? 
$$S=\sum_{n=1}^{\infty}\arctan(\frac{1}{n})$$
$$\lim_{n\to \infty} \arctan(\frac{1}{n})=0$$
So S (slowly) stops growing when n gets larger and larger. So why doesn't it converge? Shouldn't it stop growing near infinity, making it converge?

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\begin{align}
\left.\arctan\pars{1 \over n}\right\vert_{\ n\ \geq\ 1} & = {1 \over n}\,\left.1 \over \xi^{2} + 1\right\vert_{\ 0\ <\ \xi\ <\ 1/n} > {1 \over n}\,{1 \over \pars{1/n}^{2} + 1} = {n \over n^{2} + 1} > {n \over n^{2} + n^{2}} = {1 \over 2}\,{1 \over n}
\end{align}
