Prove or disprove: if $\sum_{n=1}^\infty a_n$ converges then $\sum_{n=1}^\infty a_{3n}$ converges

Question

Need help with this question:

Prove or disprove: if $\sum_{n=1}^\infty a_n$ converges then $\sum_{n=1}^\infty a_{3n}$ converges

I think it's false but I couldn't find any counter example.

Answer 1

It is false. Consider, for instance:$$-\frac12-\frac12+1-\frac14-\frac14+\frac12-\frac16-\frac16+\frac13-\cdots$$

Answer 2

$$ \begin{array}{ccccccccccccccccccccccccccc} & & & &\text{3rd} & & & & & & \text{6th} & & & & & & \text{9th} \\ & & & & \downarrow & & & & & & \downarrow & & & & & & \downarrow \\ 1 & + & 0 & - & \dfrac 1 2 & + & \dfrac 1 3 & + & 0 & - & \dfrac 1 4 & + & \dfrac 1 5 & + & 0 & - & \dfrac 1 6 & + & 0 & + & \dfrac 1 7 & - & \cdots \end{array} $$ This whole series converges to $\log_e 2\approx 0.69$ but the terms with indices that are multiples of $3$ diverges to $-\infty.$

However, in the case of series that converge absolutely, the proposed statement is true.

Answer 3

Choose a diverging sequence and intersperse terms that cancel them (f.i. $a_{n},-a_{n},0$).