Absolute Convergence and Divergence — Question 4

*Question 4 Consider $\displaystyle\sum_{n=1}^{\infty}\frac{\sin n}{n}$.

1. Prove ordinary convergence using Dirichlet’s Test.

2. Prove the absolute-value series diverges.

3. Classify the series precisely.

*Question 4 – Solution

From the finite geometric sum of $e^{in}$, $$\left|\sum_{n=1}^N e^{in}\right|\le\frac2{|1-e^i|}.$$ Taking imaginary parts shows that $\sum_{n=1}^N\sin n$ is uniformly bounded.

The sequence $1/n$ decreases to zero. Hence $\sum(\sin n)/n$ converges.

Since $0\le|\sin n|\le1$, we have $|\sin n|\ge\sin^2n$. Section 11-4 established that $\sum\sin^2n/n$ diverges; therefore $$\sum\frac{|\sin n|}{n}\ge\sum\frac{\sin^2n}{n}=\infty.$$ Thus the series is conditionally convergent.