Highly oscillatory sums

Question

Is there a way of calculating sums of (highly) oscillatory functions?

Example. How can we calculate the following series? (At least asymptotics of it.)

$$ \sum_{n=1}^{x} \frac{1}{\sin^2(1/n)} $$

Answer 1

$$\csc^2{(1/n)}=n^2+\frac13+O(1/n^2)$$ So for large $x$ your summation becomes $$\sum_{n=1}^x \left(n^2+\frac13+O(1/n^2)\right)=\frac16x(x+1)(2x+1)+\frac13x+O(1)$$