Proving/Disproving Series Transformation Properties

[daniel_i_l]

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)

Homework Equations

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.

 

[DavidWhitbeck]

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.

 

[daniel_i_l]

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".

Thanks for your help.