Consider the series
Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.
The series is conditionally convergent.
Proof. First, we recall the formula for the sine of a sum,
Therefore, we have
(Here we use that for all and for all .) This series is convergent since
and it is monotonically decreasing for .
This convergence is conditional since
But then, since and is decreasing for we have
for all . We know diverges since it is asymptotically equivalent to (i.e., ). Therefore, by the comparison test
diverges. Therefore, the convergence of the series in question is conditional