This is a question that my friend gave me $$ \sum_{r=1}^n \arctan{(r)}$$
He also told me it is a very typical question different from regular arctan questions which are solved mostly using telescoping series. He gave me a hint that it involves complex numbers.
Now, I can relate that it has to do with the argument of a complex number defined by :
$$ \ z_r = 1 + ri $$
If we take product of all such $z$ and take its argument, then we may find the sum
$$ \operatorname{Arg} \left[\prod_{r=1}^n z_r \right]$$
But I am then getting stuck in another unsolvable series.
Please help.
From comment of user: J.G and from here we have:
$$\sum_{r=1}^n \arctan{(r)}=\frac{n \pi }{2}-\Im\left(\ln \left(\frac{\Gamma ((1+i)+n)}{\Gamma (1+i)}\right)\right)$$