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|(\sin na)/(\ln10^n)\right|\to0` as
`n\to\infty` and this summand is non-increasing.

You have


`\left|(\sin na)/(\ln10^n)\right|\le\frac1{\ln10^n}=\frac1{n\ln10}\to0`

and
`\frac1{n\ln10}` is clearly strictly decreasing. This means the alternating series converges.