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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.

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

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    $\begingroup$ By "convergent" do you mean absolutely convergent, or what? $\endgroup$ – kimchi lover Dec 10 '17 at 16:32
  • $\begingroup$ @Rwitam Jama, your point it would not be useful, since the series converges $a_n$ tends to zero, so is every subsequence. $\endgroup$ – DonQuixote Dec 10 '17 at 16:32
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    $\begingroup$ 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). $\endgroup$ – lulu Dec 10 '17 at 16:33
  • $\begingroup$ @lulu If $a_{3n} = (-1)^n/n$ then $\sum a_{3n}$ converges. $\endgroup$ – zhw. Dec 10 '17 at 16:59
  • $\begingroup$ @zhw Right, of course. So make $a_{3n}=\frac 1n$, even simpler. $\endgroup$ – lulu Dec 10 '17 at 17:03
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$$1/2 + 1/2 -2/2 +1/3 +1/3 -2/3 + 1/4 + 1/4- 2/4 + 1/5 + 1/5 -2/5 + \cdots $$

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