Determine whether the following infinite series is convergent or not.

Question

Determine whether the infinite series P∞ $$\sum_{n=1}^∞ e^\frac{i\pi n}{2}\frac{1}{(n)^2}$$ is convergent or not.

*From my studies I have made the assumption: $$\sum_{n=1}^∞ e^\frac{i\pi n}{2}\frac{1}{(n)^2} = \sum_{n=1}^∞ \cos\left(n\frac{\pi}{2}\right)\frac{1}{(n)^2} + i\sum_{n=1}^∞ \sin\left(n\frac{\pi}{2}\right)\frac{1}{(n)^2}$$

*(I am not sure if this is correct)

Answer 1

$$\sum_{n=1}^{\infty}|e^{\frac{i \pi n}{2}}\frac{1}{n^2}| =\sum_{n=1}^{\infty}\frac{1}{n^2}< \infty$$

Note that $|e^{\frac{i \pi n}{2}}|=1,\forall n \in \Bbb{N}$

The series converges absolutely thus converges.