Is following sum convergent?
$$\sum_{n=1}^{\infty} \frac{Sin^2(n)}{n}$$ Integral test, Dirichlet test doesn't apply. Any idea !
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1$\sum_{n=1}^\infty \frac{\sin^2 n}{n} = \frac12 \sum_{n=1}^\infty \frac{1-\cos(2n)}{n}$, the part $\sum_{n=1}^\infty \frac{\cos(2n)}{n}$ converges by Dirichlet test. This leaves us with the part $\sum_{n=1}^\infty \frac{1}{n}$ which diverges logarithmically, so the whole thing diverges. – achille hui Oct 19 '14 at 02:15