Does the series $\sum_{n=1}^{\infty} \frac{1}{n^3 \sin^2 n}$ converge?

(Taken from http://math.stackexchange.com/questions/20555/are-there-any-series-whose-convergence-is-unknown where there are more such examples)

Does the series $\sum_{n=1}^{\infty} \frac{1}{n^3 \sin^2 n}$ converge?

(Taken from https://math.stackexchange.com/questions/20555/are-there-any-series-whose-convergence-is-unknown where there are more such examples)

Does the series $\sum_{n=1}^{\infty} \frac{1}{n^3 \sin^2 n}$ converges?

(Taken from http://math.stackexchange.com/questions/20555/are-there-any-series-whose-convergence-is-unknown where there are more such examples)

Does the series $\sum_{n=1}^{\infty} \frac{1}{n^3 \sin^2 n}$ converge?

(Taken from http://math.stackexchange.com/questions/20555/are-there-any-series-whose-convergence-is-unknown where there are more such examples)