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Showing posts from August, 2015

### The sum of all positive even integers is -1/6

Look, this is a weird result, but it is 100% trick-free, correct, provable result. I can prove to you the following statement: The sum of all positive even integers is -1/6 Relax, don't freak out, just take another sip of the wine and abstract your mind for few minutes. This will be proved. Without further due, let's get to work: What we want to prove is this: S_even = 2 + 4 + 6 + 8 + .... + oo = -1/6. First, we'll define few other sums: S1 = 1 - 1 + 1 - 1 + 1 - 1 + 1.... So S1 keeps alternating between 0 and 1. Here is how to solve it, subtract one from it and you'll end up with: 1 - S1 = 1 - (1 - 1 + 1 - 1 + 1 - 1 + 1....) getting rid of the brackets: 1 - S1 = 1 - 1 + 1 - 1 + 1... but the right side is S1 itself, hence: 1 - S1 = S1 2S1 = 1 S1 = 1/2 The second sum will be a slightly more complicated: S2 = 1 - 2 + 3 - 4 + 5 - 6 + .... To solve it, let's add S2 to itself: S2 + S2 However, to solve that addition we'll shift the botto