Convergence of $\sum\limits_{n=1}^\infty \sin^2(\pi(n + \frac{1}{n})) $?

Convergence of $\sum\limits_{n=1}^\infty \sin^2(\pi(n + \frac{1}{n})) $?

My friend was practicing for his entrance examination and one of the
problems on older exams was this one. It asks if it converges and to
explain the reasoning behind that answer. The series is:
$$\sum_{n=1}^\infty \sin^2\left(\pi(n + \tfrac{1}{n})\right)$$
We had a few ideas, but none of them seemed to be working. Hoping for some
help!