Divergent Series: Rearranging for Convergence?

[JG89]

Is it possible to re-arrange the terms of a divergent series such that the re-arranged series converges?

 

[Office_Shredder]

Yes. The alternating harmonic series gives us the way to do this in stunning fashion

1-1/2+1/3... converges. But the two series

1+1/3+1/5+1/7... and -1/2-1/4-1/6... both diverge. So here's the strategy:

Start taking positive terms until you pass the value 2. So that's 1+1/3+1/5+...+1/15. Now subtract the the first negative term, -1/2. Now add positive terms until you pass 3 (that's going to be a lot of them). Then subtract another negative term. Then add positive terms until you pass 4. Add one more negative term. Repeating this process is possible since
1) There's always another negative term to subtract
2) Since the series of positive terms diverges, we can always find enough of them to increase the value to the next integer

So as you go further along, since each negative term is less than 1, you see the partial sums get larger and larger to infinity, and this re-arrangement gives a series that diverges. To specifically do what's in the OP, you just un-rearrange and go back to the alternating harmonic series.In fact, the conditions for when this can occur