On an exercise with a series: $\sum_{n=1}^{\infty}n^2 \Big(\sin(\frac{1}{n^{\alpha}})-\frac{1}{n^{\alpha}+1}\Big)$ The series in question is the following:
$$\sum_{n=1}^{\infty}n^2 \Big(\sin(\frac{1}{n^{\alpha}})-\frac{1}{n^{\alpha}+1}\Big)$$
I want to study for which $\alpha >0$ does the series converge.
I tried the root test to not much avail, after I tried expanding with taylor the sine function (by the way I can taylor expand a function even if the input are only ration numbers of the form $\frac{1}{n^{\alpha}}$ right?) but I still haven't got anywhere.
We have not done the integral test, is that the way to go?

Hint: Write $\displaystyle  \sin(x)-\frac{x}{1+x}=x^2+O(x^3)$.
