Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

Question

I need help with this series. Series of sin(9n)/3^n from n=1 to infinity. Thanks.

Answer 1

Hi Ab,

 

If ∑a_n is your series, notice |sin (9n)| ≤ 1 because the range of the sin function is [-1, 1]. Therefore, ∑|a_n| ≤ ∑(1/3)ⁿ. The second series is a geometric series with common ratio |r| < 1 so the second series is convergent. By the comparison test, ∑a_n is absolutely convergent.