Let \(w_1,w_2,\ldots,w_n\) be positive real numbers such that \( \sum_{i=1}^n w_i=1\). Prove that if \(x_1,x_2,\ldots,x_n\in [0,\pi]\), then \[ \sin \left(\prod_{i=1}^n x_i^{w_i} \right) \ge \prod_{i=1}^n (\sin x_i)^{w_i}.\]
Let X be the set of all postive real numbers c such that \[\frac{\prod_{k=1}^{n-1} \sin\left( \frac{k \pi}{2n}\right)}{c^n} \] converges as n goes to infinity. Find the infimum of X.
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!