Proof the sum of all positive integers is equal to negative one over twelve
Let a = 1 - 1 + 1 - 1 + 1 - 1 ...
2a = 1 - 1 + 1 - 1 + 1 - 1 ...
+ 1 - 1 + 1 - 1 + 1 ...
= 1
a = 1/2
Let b = 1 - 2 + 3 - 4 + 5 - 6 ...
2b = 1 - 2 + 3 - 4 + 5 - 6 ...
+ 1 - 2 + 3 - 4 + 5 ...
= 1 - 1 + 1 - 1 + 1 - 1 ...
= 1/2
b = 1/4
Let c = 1 + 2 + 3 + 4 + 5 + 6 ...
c - 4c = 1 + 2 + 3 + 4 + 5 + 6 ...
- 4 - 8 - 12 ...
-3c = 1 - 2 + 3 - 4 + 5 - 6 ...
= 1/4
3c = -1/4
c = -1/12
Therefore the sum of all positive integers (1,2,3,4,5,6...) is equal to negative one over twelve. Q.E.D.