Conditionally convergent series

If u_1+u_2+...=s where series converge conditionally and t>s, then there exists a derangement of the series, which converges to t and which leaves all positive numbers of the series on their places. How to prove it?

 

Do this in a similar way that you show that you can get a
rearrangement of a conditionally convergent series to sum to any value.

 

I assume that" leaves all positive numbers of the series on their places" means that the positive numbers are in the same order as in the original series.

Make separate lists of the positive and negative numbers. Add from the positive numbers, in order, until the first time the sum is larger than t (Since, in order to be 'conditionally convergent' rather than 'absolutely convergent', the series of all positive terms must not converge, this will eventually happen). Now start adding numbers from the list of negative numbers until teh sum is back below t (again, the series of all negative terms must not converge so this will happen). Now go back to adding positive numbers until you are back above t. Since, in order to converge, the terms of the series are going to 0, we go "over" or "below" t by smaller and smaller amounts each time. That shows that the series converges to t.

 

Such a conditionally convergent real series can be MADE to converge to an arbitrary real
number t, but need not do so unless suitably rearranged.