Give examples of following series :

Question

Give an example of the following series :

(i) $\sum\limits_{n=1}^{\infty} a_n$ is convergent but $\sum\limits_{n=1}^{\infty} a_{3n}$ is divergent
(ii) $\sum\limits_{n=1}^{\infty} a_n$ is divergent but $\sum\limits_{n=1}^{\infty} a_{3n}$ is convergent

I have tried to find such example, but failed. If we take $a_n=\frac{1}{n}$, then $\sum\limits_{n=1}^{\infty} a_{n}$ is divergent but what can we say about $\sum\limits_{n=1}^{\infty} a_{3n}$?

Any hints will be appreciated.

Answer 1

For the first case, let $$a_{3k}=\frac1k\qquad a_{3k-1}=a_{3k-2}=-\frac1{2k}$$ for every positive integer $k$.

And for the second case: $$a_n=\sin\frac{n\pi}3$$