# Series question

Gold Member

## Homework Statement

Prove or disprove:
There exist series $$\sum a_n$$ and $$\sum b_n$$ so that:
1) you can get $$b_n$$ by rearranging the elements of $$a_n$$
2) $$\sum b_n = 2 + \sum a_n$$
3) $$\sum |b_n| = 2 \sum a_n$$
(all the series converge to finate values)

## The Attempt at a Solution

From (1) I know that $$\sum |b_n| = \sum |a_n|$$ but I can't see how can to continue from here, can someone point me in the right direction?
Thanks.

You can't say that all series mentioned converge, because if $$\sum |b_n|$$ converges then the series is absolutely convergent which means that any reordering of the original series $$\sum b_n$$ converges to the same value, which implies that $$\sum a_n = \sum b_n$$ by virtue of (1). That means that (1) contradicts (2). So I have to say that it's impossible. But I could be wrong.

Gold Member
You can't say that all series mentioned converge, because if $$\sum |b_n|$$ converges then the series is absolutely convergent which means that any reordering of the original series $$\sum b_n$$ converges to the same value, which implies that $$\sum a_n = \sum b_n$$ by virtue of (1).
How do you know that rearranging the elements in an absolutly converging series doesn't change their value?

EDIT: Oh, it's easy to see that that's true if you split up each of the series into positive and negative "sub-series".