Evaluate the sum \[ \sum_{n=1}^{\infty} \frac{n \sin n}{1+n^2}. \]

(UPDATED: 2011.2.18) I have fixed a typo in the formula. Initially the following formula \[ \sum_{n=1}^{\infty} \frac{\sin n}{1+n^2}\] was posted but it does not seem to have a closed form answer. I’m sincerely sorry!