Give an example of a divergent series $\sum a_n$ such that $\sum a_{3n}$ is convergent.

Question

Give an example of a divergent series $\sum a_n$ such that $\sum a_{3n}$ is convergent.

I am not able to find such example. Please help.

Answer 1

$$a_n=\sin(\frac{n\pi}3)$$

Clearly, $$\sum_{n=1}^\infty a_n$$ is analogous to $1-1+1-1+1-\cdots$ which does not converge but $$a_{3n}=0$$ for all $n$ and $$\sum 0=0$$ (or rigorously the sequence of partial sum is a sequence of zeroes, thus the limit of the infinite sum is zero).