[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.