The wikiepedia entry is interesting, but it involves some math that I don't quite understand. Apparently the other reddit readers don't understand it either, since one of the top comments there says, "Really, there's no sum (the sum is undefined). 1/4 is just some number that mathematicians can use to compare sequences like that."
The answer 1/4 is a little less arbitrary than that!
Instead, try simply doing the arithmetic:
1 = 1
1 - 2 = -1
1 - 2 + 3 = 2
1 - 2 + 3 - 4 = -2
1 - 2 + 3 - 4 + 5 = 3
1 - 2 + 3 - 4 + 5 - 6 = -3
See the pattern? It's 1, -1, 2, -2, 3, -3, 4, -4, 5, -5...
What happens if we plot those points on a graph and then draw separate lines for the positive and negative numbers? (first point at position [0,1], the second at [1,-1], third at [2,2], fourth at [3,-2], etc)
The line of positive numbers is defined by the equation y = x/2 + 1
The line of negative numbers is defined by the equation y = -x/2 - 0.5
Notice that the two lines intersect at y = 0.25, which also happens to define the line midway between the other two lines (since they have slopes of +1/2 and -1/2).
It seems that if we "average" the two equations, like so:
(y + y) / 2 = ((x/2 + 1) + (-x/2 - 0.5)) / 2
simplified:
y = (x/2 - x/2 + 1 - 0.5) / 2
simplified:
y = 1/4
Yay!
I don't know if a real mathematician would approve of my method, but at least it's easy to understand.
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