# 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.

Related Calculus and Beyond Homework Help News on Phys.org
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".