Determine the radius of convergence of the power series:

Test for convergence at the boundary points if is finite.

If then unless . Thus, the series does not converge for if .

If then we apply Dirichlet’s test where

is bounded for any (i.e., the partial sums are bounded) and the terms are monotonically decreasing with . Hence, the series converges for , which implies if .

If then for all so the series converges for all which implies .

This | sine | series diverges though

You could have said it’s going to be less or equal to |z|^n which converges if |z| 1 but this is no problem since, as I said above, | sin(ax) | diverges when |z| = 1, so… yeah

Btw, the sine series diverges because the limit of the absolute value doesn’t go to zero (see theorem 10.6)