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

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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.

## marked as duplicate by Micah, Namaste, Dando18, B. Goddard, StrantsMay 31 '18 at 23:27

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• By "convergent" do you mean absolutely convergent, or what? – kimchi lover Dec 10 '17 at 16:32
• @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
• @lulu If $a_{3n} = (-1)^n/n$ then $\sum a_{3n}$ converges. – zhw. Dec 10 '17 at 16:59
• @zhw Right, of course. So make $a_{3n}=\frac 1n$, even simpler. – lulu Dec 10 '17 at 17:03

## 1 Answer

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