Does $\sum_{n=1}^\infty {\frac{e^{i{\frac{\pi}{n}}}}{\sqrt n}}$ converge?

Question

I have $$\sum_{n=1}^\infty {\frac{e^{i{\frac{\pi}{n}}}}{\sqrt n}}$$ First I check whether it converges absolutely (in which case in converges). $$ \sum_{n=1}^\infty {\frac{e^{i{\frac{\pi}{n}}}}{\sqrt n}} = \sum_{n=1}^\infty {\frac{1}{\sqrt n}} \times |e^{i{\frac{\pi}{n}}}| = \sum_{n=1}^\infty {\frac{1}{\sqrt n}} \times 1 $$ Since ${\frac{1}{\sqrt n}}$ converges then I'd say the whole series converges. But the answer is that it doesn't converge. What am I doing wrong?

Answer 1

The series $\sum_{n=1}^\infty\frac1{\sqrt n}$ diverges. Therefore, your argument is flawed.

On the other hand$$(\forall n\in\mathbb{N}):\operatorname{Re}\left(\frac{e^{i\frac\pi n}}{\sqrt n}\right)=\frac{\cos\left(\frac\pi n\right)}{\sqrt n}\geqslant\frac1{2\sqrt n}$$if $n$ is large enough. Therefore, yes, your series diverges.