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

Question

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

I am trying to construct an example such that $\sum a_n$ is convergent but $\lim a_{3n}$ is not zero. But I could not find such example. Give me some hints.

@Rwitam Jama, your point it would not be useful, since the series converges $a_n$ tends to zero, so is every subsequence. — DonQuixote, Dec 10 '17 at 16:32
Hint: start with the alternating harmonic sequence, so $a_{3n}=\frac {(-1)^n}n$. That diverges. Now add terms to make the full sequence converge (conditionally). — lulu, Dec 10 '17 at 16:33
@zhw Right, of course. So make $a_{3n}=\frac 1n$, even simpler. — lulu, Dec 10 '17 at 17:03

score 1 · Answer 1 · answered Dec 10 '17 at 16:58

1

$$1/2 + 1/2 -2/2 +1/3 +1/3 -2/3 + 1/4 + 1/4- 2/4 + 1/5 + 1/5 -2/5 + \cdots $$

answered Dec 10 '17 at 16:58

zhw.

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

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