# Closed form for $\sum_{k=1}^{n}\frac{\sin\frac{\pi(k-n)}{2k+1}}{\sin\frac{\pi(k-n-h)}{2k+1}}$?

I am trying to find the closed form for

$$\sum_{k=1}^{n}\frac{\sin\frac{\pi(k-n)}{2k+1}}{\sin\frac{\pi(k-n-h)}{2k+1}}$$

where $h$ is an arbitrary integer.

Any hints will be appreciated, thank you.

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Generalize in that sense? –  Andrew Jul 15 '11 at 8:41
I trust that this is not for some sort of exam or a homework assignment? –  r.g. Jul 15 '11 at 8:56
It strikes me as quite unusual, since the denominator $2k+1$ varies within the sum. –  GEdgar Jul 15 '11 at 13:16
This series is very important. if someone solve this problem, i`ll show you. –  4545454545SI Jul 15 '11 at 15:25
@ks0830: It's reassuring to know that you consider it important (hopefully so, otherwise why would you ask?). However, I'd love to see some motivation before thinking about this problem. –  t.b. Jul 15 '11 at 15:33