Convergence of $\sum^\infty_{n=1}\Big(1+\frac12+\ldots+\frac1n\big)\frac{\sin nx}{n}$

calculussequences-and-series

Determine all values of $x$ for which the series
$$
\sum^\infty_{n=1}\Big(1+\frac12+\ldots+\frac1n\big)\frac{\sin nx}{n}
$$
converges.

I tried to use Dirichlet's criteria an noticing that $\frac{1}{n}(1+\ldots + \frac1n)\sim \frac{\log n}{n}$ but I got stuck. Any additional hints will be appreciated.

Best Answer

Some hints: If $\sin\frac x2 = 0,$ then the sum is obviously zero. So assume otherwise. Then, show two things:

  1. $b_n = \sin x+\sin(2x)+...+\sin(nx)$ is uniformly bounded over all $n$ for given $x.$ The sum actually has a nice closed formula and one way to deduce that would be to multiply and divide by $\sin\frac x2$ and then use trig identities.

  2. $a_n = \frac 1n\left(1+\frac 12 +\frac 13+\dots + \frac 1n\right)$ is monotonically decreasing to zero.

Then, you will be ready to use the Dirichlet Criterion.

EDIT: Monotonicity of $a_n.$ Let $H_n = 1+\frac 12+\dots \frac 1n$ be the harmonic sum. Then $a_n > a_{n+1}$ is equivalent to: $$\frac 1nH_n > \frac{1}{n+1}H_{n+1} = \frac{1}{n+1}\left(H_n+\frac{1}{n+1}\right).$$ But the above is equivalent to, above some simplification : $$H_n>\frac{n}{n+1},$$ which is trivial since $H_n\geq 1.$

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