2
$\begingroup$

Let $a_1, a_2, a_3,...$ be a sequence of non-negative real numbers, let $S_1, S_2, S_3,...$ be a sequence (finite or infinite) of disjoint nonempty sets of natural numbers whose union is $\{1,2,3,...\}$ and suppose that for each $i$ such that $S_i$ is infinite the series $\sum_{n\in S_i}a_n$ converges and that if the number of sets $S_1, S_2, S_3,...$ is infinite then the series $\sum_{i=1}^\infty(\sum_{n\in S_i} a_n)$ converges.

Prove that the series $\sum^\infty_{n=1} a_n$ converges.

$\endgroup$
  • $\begingroup$ I edited your post do you mind checking if it is as intended? $\endgroup$ – azarel Sep 24 '13 at 17:35
  • $\begingroup$ @azarel yes it is perfect thank you! I really need to learn how to do that for myself... $\endgroup$ – Stephanie Wayne Sep 25 '13 at 22:41
1
$\begingroup$

It suffices to show that the sequence $s_m$ of partial sums is bounded ($s_m=\sum_{n=1}^m$). Let $M=\sum_{i=1}^\infty(\sum_{n\in S_i}a_n)$. Now, let us show that $s_m\leq M$ for all $n\in\mathbb N$. For $m\in\mathbb N$, pick $N$ so that $\{a_1,...,a_m\}\subseteq \bigcup_{i=1}^N S_i$. It follows that $$s_m\leq\sum_{i=1}^N(\sum_{n\in S_i}a_N)\leq M$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.