Question

Prove that the following series is convergent for all 𝑟 ∈ ℝ

Σ (1 + 1/2 + ... + 1/n) (sin (nr) / n).

Answer 1

let:

$a_n=\frac{Σ ( 1/n)}{ n}|sin(nr)|$

$b_n=\frac{Σ ( 1/n)}{ n}$

$|sin (nr)|\le 1$ , then:

$a_n\le b_n$

also:

$b_n\le \sum (1/n^2)$

So, since $\sum (1/n^2)$ converges, $b_n=\frac{Σ ( 1/n)}{ n}$ converges as well

so, $\sum b_n$ converges

then, $\sum a_n$ converges

so, series $\sum (1+1/2+...+1/n)\frac{sin(nr)}{n}$ converges