If the series $\sum_{n=1}^\infty a_n$ converges absolutely

and $a_{n_k}$ is sub sequence of $a_n$

does the series $\sum_{n=1}^\infty a_{n_k}$ also always converge ?

Since I failed to found any counter-example, and the only connection between absolutely convergence of $\sum_{n=1}^\infty a_n$ and convergence of $\sum_{n=1}^\infty a_{n_k}$ in my textbook is this theorem: the series $\sum_{n=1}^\infty a_n$ converges absolutely iif the series of positive and negative members of $a_n$ converges, I concluded that my proof must be somehow connected to this theorem.

In this stage I arrived to the dead-end.

Could you please give me some hint how to deal with this question ?



Yes, it does. Since $\{n_k : k \in \mathbb{N}\} \subseteq \{n : n \in \mathbb{N}\}$, we see that

$$\sum_k \left|a_{n_k}\right| \le \sum_n |a_n| < \infty$$

So the relevant sum is, in fact, absolutely convergent.

  • $\begingroup$ Could you please explain why $\sum_k |a_{n_k}|\le \sum_n |a_n| $? $\endgroup$
    – user97484
    Dec 1 '13 at 9:19
  • 2
    $\begingroup$ @user97484 You're adding up fewer positive things, so you get a smaller answer. $\endgroup$
    – user61527
    Dec 1 '13 at 9:20

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