[ColinWright]

  > A1 and A2 have different limits. It will be clear
  > if the upper limit is changed to something other
  > than infinity.

While it might be true that A1 and A2 have different limits, the reason you give is not sufficient to prove it. Consider the following:

  1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

and

  1/2 + 1/8 + 1 + 1/32 + 1/128 + 1/4 + ...

In this second case I'm taking two odd powers then an even power, and so on.

If your argument were valid then these would have different limits, because if you cut them both off after some number of terms the totals will always be different. However, these two series in fact have the same limit.

And in fact A1 and A2 have the same limit, so that's not where the problem lies.

[slevin063]

>For A1 and A2 to be equal, A2 will have limits x=1 to 5.

I should have added 'For A1 and A2 to be equal' at the start. If you want them to be equal, they need to have different limits and if they have same limits, they are not equal.

[ColinWright]

I'm finding that completely impossible to parse.

In particular:

    If you want them to be equal, they need
    to have different limits and if they have
    same limits, they are not equal.

That seems to be completely wrong. Consider these:

    sum_{x=0}^oo (-1)^x (1/x)

and

    sum_{y=0}^oo [ 1/(2y) - 1/(2y+1) ]

These are equal, and yet replacing the "oo" with n always gives partial sums that are different. So I guess I just don't understand what you are trying to say. Perhaps you could be more complete and explicit.