Recall that [tex]|\sin x|\le1[/tex], so this is an alternating series.
The series will then converge iff [tex]\left|\dfrac{\sin na}{\ln10^n}\right|\to0[/tex] as [tex]n\to\infty[/tex] and this summand is non-increasing.
You have
[tex]\left|\dfrac{\sin na}{\ln10^n}\right|\le\dfrac1{\ln10^n}=\dfrac1{n\ln10}\to0[/tex]
and [tex]\dfrac1{n\ln10}[/tex] is clearly strictly decreasing. This means the alternating series converges.
