Range of convergence of series Find all real value $a$ for which the series $$ \sum_{n=1}^{\infty}
{(\frac{1}{n} -\sin(\frac{1}{n}))^a}$$ convergent.
I tried using ratio test, logarithmic test, etc. But I could not find it.

Since $\lim\limits_{n\rightarrow +\infty}\frac{1}{n}=0$, you have
$$ \frac{1}{n}-\sin\left(\frac{1}{n}\right)\underset{n\rightarrow +\infty}{\sim}\frac{1}{6n^3}$$
Thus
$$ \left(\frac{1}{n}-\sin\left(\frac{1}{n}\right)\right)^a\underset{n\rightarrow +\infty}{\sim}\frac{1}{6^an^{3a}} $$
and the series converges iff $3a>1$ that is to say $a>\frac{1}{3}$.
