>> s = 1-1+1-1+1-1+1-...
>> s = (1-1)+(1-1)+(1-1)+(1-1)+...
>> s = 0+0+0+...
>> s = 0
>
>> Then the "..." in the 3rd (and perhaps 2nd) line makes the assumption
>> that there are an even number of terms
>
> No, it doesn't. It simply makes the assumption that you can choose
> each odd-placed and even-placed number in the series (which is true)
> and sum them together (which is also true). This can be done forever.

Yes, I just realised that you could also write s as:

s =+1+1+1+1+1+...
    -1-1-1-1-1-...
   = 0+0+0+0+0+...

Which makes no such assumptions.

But then you could probably just as validly (which might not be valid at 
all) write s as:

s = +1+1+1+1+1+1+...
         -1-1-1-1-...
   =  1+1+0+0+0+0+...

So essentially you could "prove" any value you like for s.

Funnily enough if you use the standard formula for the infinite sum of 
geometric progressions, you also get 1/2.

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