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Let $\sum_{n=1}^{\infty} a_n$ converge. Let $(n_k)_{k=1}^{\infty}$ be a subsequence of the sequence of positive integers. For each k, define:

$$b_k = a_{n_{k-1}+1} + ...+ a_{n_k}$$ where $n_0 = 0$.

Prove $\sum_{n=1}^{\infty} b_k$ converges and that $\sum_{n=1}^{\infty} a_n = \sum_{k=1}^{\infty} b_k$.

This is the question I am looking at. I know that each subsequence of a convergent sequence also converges, and therefore because if a series converges, the sequence must also converge.

I guess I'm just struggling notationally with this. I'm not really sure what $b_k$ is defining. I'm not looking for an answer (besides, this problem will not be graded, it is for practice), but a little help in the right direction would be greatly appreciated.

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1 Answer 1

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You've got precisely the right idea -- using the fact that any subsequence of a convergent sequence converges to the same thing.

Hints:

  1. What does it really mean for $\sum a_n$ to converge?

  2. What does the sequence of partial sums of the $b_k$ look like?

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  • $\begingroup$ The sum converges when for all $\epsilon > 0$ there exists N such that for $n \geq N$ and $p \geq 0$ then $$|a_n + a_{n+1}+ ...+ a_{n+{p_1}}| < \epsilon $$ $\endgroup$
    – o's1234
    Commented Oct 18, 2020 at 22:10
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    $\begingroup$ @o's1234 So "$\sum_{n\in\mathbb{N}} a_n$ converges" if and only if the sequence of partial sums $(\sum_{n=1}^N a_n)_{N\in\mathbb{N}}$ converges. Is there a similarity between that sequence of partial sums and the sequence $(\sum_{n=1}^M b_n)_{M\in\mathbb{N}}$? If you're not sure, it may be helpful to try writing out the terms of both sequences in an example, e.g. with $(N,M)=(n_1,1)$ or $(N,M)=(n_2,2)$ $\endgroup$
    – Zim
    Commented Oct 19, 2020 at 2:28
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    $\begingroup$ @o's1234 Well the first line you wrote should be the $n_1$th partial sum of the $a_n$'s, right? So that is not actually $a_1$ -- it is $\sum_{k=1}^{n_1}a_k=a_1 + \cdots + a_{n_1}$. But wait, this is the same as $b_1$! $\endgroup$
    – Zim
    Commented Oct 19, 2020 at 16:47
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    $\begingroup$ @o's1234 can you find the similarity between $\sum_{k=1}^{n_2}a_k$ and $b_1+b_2$? You're on the right track -- the $b_n$s are not quite the same as the partial sums of $a_n$. However, once we start adding up the $b_n$s I think you'll find a pattern :) $\endgroup$
    – Zim
    Commented Oct 19, 2020 at 16:47
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    $\begingroup$ @o's1234 yep, pretty much! It's just a matter of finding which indices of the $a_n$ sums match up with the $b_n$ sums. Then you can say that the sequence of partial sums of $b_n$s is a subsequence of the sequence of partial sums of $a_n$s $\endgroup$
    – Zim
    Commented Oct 19, 2020 at 17:04

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