Absolute Convergence and Divergence — Question 4

Question 4 – Solution

Step 1: Bound the numerator partial sums.

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.

Step 2: Apply Dirichlet’s Test.

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

Step 3: Rule out absolute convergence.

Since $0\le|\sin n|\le1$, we have $|\sin n|\ge\sin^2n$. Section 4-4 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.