Convergence of $\sum_{n=1}^{\infty}\frac{1}{3^n\ \sin(n)}$
Does this series converge? Root test and ratio test are inconclusive.
Solution 1:
Using the same arguments as in the two answers of Does $\sum_{n=1}^\infty \frac{1}{n! \sin(n)}$ diverge or converge? we can compare the series $\sum^\infty_{n=1}\left|\frac{1}{3^n\sin n}\right|$ to $\sum^\infty_{n=1}\frac{n^7}{3^n}$, which certainly converges by the ratio test, so that the original series converges (absolutely).