Ratio Test:

(i) If , then the series is  is absolutely convergent.

(ii) If  or , then the series is  is divergent.

(iii) If , then the ratio test is inconclusive.

The series is .

Consider .

Apply ratio test:

.

The sine function is bound between and , therefore for all values of .

.

Multiply both sides of the inequality by .

.

Theorem 4:

The geometric series is convergent if .

The series is smaller than the convergent series, hence it must be converge by the comparision test.

Thus by theorem 4, is convergent.

Therefore by the comparision test,   is convergent.

The series is absolutely convergent.