19: The usual sine approximation from above by 1 is useless, since the series with terms 1 diverges and is therefore useless as an upper estimate. Of more interest is the investigation of behaviour of terms for large k. You should get
This tells us that there is a chance for convergence, but nothing more. The crucial question is how fast the terms go to zero. What do you know about values of sine for small arguments? Some hints may be found in Approximation of functions by tangent lines in Derivatives - Theory - Applications, or Taylor series in Series - Theory - Series of functions. You should get
This looks like an invitation to limit comparison. Justify this estimate and then check on the convergence of the test series.