April 2007 (UTC) 

 :::Rosa, I think this might help.  In mathematics, if you can define a set of rules, that's consistent with itself, then you can work with those rules.  According to the definition of "sum" and "equals" that most of us are used to, it makes no sense to say that 1 - 2 + 3 - 4 . . .  equals anything. 
 :::However, there are other, somewhat more generous, ways of defining "sum" and "equals", that don't contradict our ordinary, finite, arithmetic, but which produce some additional surprising results with infinite sums.  There are definitions and sets of rules under which experssions like the one in this article can make sense, and add up to a particular number.  Within those rulesets, the sum is, oddly enough, 1/4. 
 :::I think one good way to see how that could possibly work is to consider this: if you take the series (1 - 2 + 3 - 4 . . .), and write it down 4 times in just the right way, you can see how all the positive terms and all the negative terms cancel out, except for one of the initial 1's.  Thus, four copies of the series add up to 1, so each one must equal 1/4. - 01:30, 19