Find all x $\in$ R for which $\sum_{n=1}^\infty(1+\frac 12+...+\frac 1n)\frac {\sin nx} n$ converges.

Question

Find all $x \in \Bbb{R}$ for which $$\sum_{n=1}^\infty\Big(1+\frac 12+...+\frac 1n\Big)\frac {\sin nx} n$$ converges.

In the beginning, I want to apply Dirichlet Test on it.

But I am still not sure that I've found all $x \in \Bbb{R}$.

Answer 1

You can write the series as $\sum_{n=1}^\infty a_n \sin nx$ where $a_n = (1 + \cdots + \frac 1n) \frac 1n$.

Obviously $$1 + \cdots + \frac 1n > \frac{n}{n+1}$$ so that $$(n+1) \left( 1 + \cdots + \frac 1n \right) > n \left( 1 + \cdots + \frac 1n \right) + \frac n{n+1} = n \left( 1 + \cdots + \frac{1}{n+1} \right)$$ which rearranges to $a_{n+1} < a_n$. Moreover $a_n \to 0$ because $0 \le a_n \le \frac{ \log(n+1)}{n}$ using the usual comparison with the area under the graph of $y = \frac 1x$.

Since $\{\sin nx\}$ has bounded partial sums and $\{a_n\}$ decreases to zero the series converges for all $x$ by the Dirichlet test.