TITLE: Evaluate $ \sum_{n=1}^{\infty} \frac{\sin \ n}{ n } $ using the fourier series
QUESTION [10 upvotes]: I am a beginner with Fourier series and I have to evaluate the sum 
$$\sum_{n  =1}^{\infty}{\sin\left(n\right) \over n}$$
I don't know which function I have to take to evaluate the fourier series ...
Someone can give me a hint ?
Thanks in advance!

REPLY [2 votes]: By the way, note that $\frac{\sin n}{n}$ is exactly the n-th Fourier coefficient of the function 
$
 \sqrt{\frac{\pi}{2}}\chi_{[-1,1]}(x).
$
Since $\chi_{[-1,1]}(x)$ has a compact support, one can use the Poisson formula: 
$$
    \sum_{n\in\mathbb{Z}} \widehat{\chi_{[-1,1]}}(n) = \sqrt{2\pi} \sum_{n \in \mathbb{Z}} \chi_{[-1,1]}(2\pi n) = \sqrt{2\pi},
$$
and get
$$
   \sum_{n\in\mathbb{Z}} \frac{\sin n}{n} = \pi.
$$