Question

Convergence of the series sin(na)/ln(10)^n

Answer 1

Recall that $|\sin x|\le1$, so this is an alternating series.

The series will then converge iff $\left|\dfrac{\sin na}{\ln10^n}\right|\to0$ as $n\to\infty$ and this summand is non-increasing.

You have

$\left|\dfrac{\sin na}{\ln10^n}\right|\le\dfrac1{\ln10^n}=\dfrac1{n\ln10}\to0$

and $\dfrac1{n\ln10}$ is clearly strictly decreasing. This means the alternating series converges.