Q: Complex and recursive sequences Study the convergences of the following sequences:$$\sum_{n=0}^{\infty} \frac{i^{n}}{(n+1)^{2}} $$
and $$\sum_{n=0}^{\infty} a_{n} $$
Where $a_{n}$ is defined recursively by: $$a_{0} = 1+i $$ $$a_{n+1} = \frac{(2+3i)n}{7+(5i)n^{2}}a_{n} $$
I don't know how to do this two excercises, if anyone can help I really appreciate.

A: Okay, since you've already done the first one by absolute convergence, you might try that for the second series as well. 
Again, since the expressions are ugly enough that we don't really want to do much with them, we look at the absolutes. We have $\frac{a_{n+1}}{a_n}=\frac{(2+3i)n}{7+(5i)n^2}$. 
Now, if we imagine we had absolutes here. Then we'd get $|\frac{a_{n+1}}{a_n}|=|\frac{(2+3i)n}{7+(5i)n^2}|\geq|\frac{\sqrt{5}n}{5n^2-7}|>\frac{\sqrt{5}n}{4n^2}=\frac{\sqrt{5}}{4n}$ for nearly all $n$ - provided I didn't miscalculate somewhere along the way, I leave the reasoning to you.
Once you're here, I'll only say ratio test...
