For which values of a does the series $\sum_{j=1}^{\infty} \frac{\sin(\frac{1}{j})^a}{j}$ converges?

Question

For which values of a does the series $\sum_{j=1}^{\infty} \frac{\sin\bigl(\tfrac{1}{j}\bigr)^a}{j}$ converges?

Answer 1

Hint:

Use equivalence: $\;\sin u\sim_0 u$, so $$ \frac{\sin\bigl(\tfrac{1}{j}\bigr)^a}{j}\sim_{j\to\infty}\frac1{j^{a+1}}.$$

Can you proceed?