Convergence of a sinusoidal infinite series from Borwein, Bailey, and Girgensohn

Borwein, Bailey, and Girgensohn (2004) asked whether the following infinite series converges: the sum of $(\frac{2}{3} + \frac{1}{3} \sin n)^n / n$ over all positive integers $n$. We prove that their series converges. The proof uses the irrationality measure of $\pi$.