1) $\displaystyle \displaystyle\sum_{n=1}^{\infty}{a_n}$ is divergent.

Proove that $\displaystyle \displaystyle\sum_{n=1}^{\infty}\frac{a_n}{1+n*a_n }$ is also divergent.

2) $\displaystyle a_n \geq a_{n+1} \textgreater 0$ and $\displaystyle \displaystyle\sum_{n=1}^{\infty}{a_n}$ is convergent.

Proove that: $\displaystyle \lim\limits_{n \rightarrow \infty}{n*a_n}=0$

I am a third year student at Computer Science, therefore I have some strong basics. Any help, direction where to look is appreciated. Thx

LE:

3) $\displaystyle

\sum_{i=0}^\infty \arctan\frac{1}{n^2+n+1}=?

$