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)