$\sum_{n=1}^{\infty} {(1-\cos(\sin 1/n))}^{w}$ with $w$ as parameter Let $f(x)=(1-\cos(\sin x))$; $a_n=f(1/n)$ for $n\in\mathbb{N}$ For which $w>0$ series $$\sum_{n=1}^{\infty} {a_n}^{w}$$ converge? I haven't got a slicest idea how to check that, absolutely none (besides Cauchy's criterion but that seems to be unwise here). Please guys, help.

To understand the behavior of $a_n$, notice that $$ f(x)=\frac{x^2}{2}+O(x^4) $$ as $x \to 0$. Hence $a_n \approx \frac{1}{2n^2}$ as $n \gg 1$. Therefore you can compare $a_n^w$ to $n^{-2w}$ when $n \gg 1$.