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?

For values of $x$ near $0$, $\arctan(x)$ ~ $x$.
IF you know calculus, this is because the rate of change of $\arctan(x)$, $\displaystyle \frac{1}{1+x^2}$, approaches $1$,the rate of change of $x$, as $x$ approaches $0$.
It is a well known fact that the harmonic series or $\displaystyle \frac{1}{x}$, that is $1+\frac{1}{2}+\frac{1}{3}...$ does not converge.
For $\arctan{\frac{1}{x}}$, as $x$ gets bigger, this series slowly starts to become the harmonic series, which diverges.
