I'm not sure what you mean by roots in the exponent of u. Can you clarify?

[Dodobird]

Examine for which u \in \mathbb R the series \sum\limits_{n=1}^\infty \frac {(1+(-1)^n)^n}{n^2} |u|^{\sqrt{n}(\sqrt{n+1})}
converges.

What I found out so far: (1+(-1)^n) alternates between [0;2], that means that the whole series becomes zero for the even n. The interesting part are the odd n but what role plays u. I´m still a bit confused with the roots in the exponent of u

Thanks...;)

 

[haruspex]

that means that the whole series becomes zero for the even n.

You mean, the even n terms vanish, right? That being so, can you rewrite the series in a simpler form, preferably in a way that has all terms positive? Then, what tests do you know for convergence of series?