Strategy for Series — Question 7

*Question 7 Classify $\displaystyle\sum_{n=1}^{\infty}\frac{\sin n}{n}$ as absolutely convergent, conditionally convergent, or divergent. Use bounded partial sums for the sine factor.

*Question 7 – Solution

The finite identity $$\sum_{n=1}^N\sin n=\frac{\sin(N/2)\sin((N+1)/2)}{\sin(1/2)}$$ shows that $\left|\sum_{n=1}^N\sin n\right|\le1/|\sin(1/2)|$.

The sequence $1/n$ decreases to $0$, and the partial sums of $\sin n$ are bounded. Hence $\sum \sin n/n$ converges.

Since $|\sin n|\ge\sin^2 n$, $$\sum\frac{|\sin n|}{n}\ge\sum\frac{\sin^2 n}{n}=\frac12\sum\frac1n-\frac12\sum\frac{\cos(2n)}n.$$ The cosine series converges by Dirichlet’s Test while the harmonic series diverges, so the absolute-value series diverges.

The series is conditionally convergent.