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 bottom sum to the right by one - it doesn't really matter whether we do it or not because we'll still end up adding up all their terms, so here we go:

S2 + S2 =

1 - 2 + 3 - 4 + 5 - 6 + .... +
     1 - 2 + 3 - 4 + 5 - 6 + ....
-----------------------------------
1 - 1 + 1 - 1 + 1 - 1 + .....

Hence,

S2 + S2 = S1
2S2 = 1/2
S2 = 1/4

Now, we'll define S3 as the sum of all the positive integers, in other words:

S3 = 1 + 2 + 3 + 4 + 5 + .... + oo

To solve it, we'll subtract S3 from S2:

S3 - S2 =

 1 + 2 + 3 + 4 + 5 + ....
-1 + 2 - 3 + 4 - 5 +....
------------------------------------
0 + 4 + 0 + 8 + 0 + 12....

Hence

S3 - S2 = 4 + 8 + 12 + 16 + 20 + ...
S3 - S2 = 4 * (1 + 2 + 3 + 4 + ....)

but the expression in brackets is S3 itself (right?), hence

S3 - S2 = 4S3
3S3 = -S2
3S3 = -1/4
S3 = -1/12

Ok, almost there. Now we'll define two sums:

S_even = 2 + 4 + 6 + 8 + 10 + .... (Btw, S_even is what we're looking for!)
S_odd = 1 + 3 + 5 + 7 + 9 + ....

It is clear that this next equation holds true:

S_even + S_odd = S3, or
S_even + S_odd = -1/12

We now need another equation. What happens when we subtract S_even from S_odd?

S_even - S_odd = 2 + 4 + 6 + 8 + 10 + ... - (1 + 3 + 5 + 7 + 9 + ....)

Getting rid of the brackets and re-arranging, and we have:

S_even - S_odd = -1 + 2 - 3 + 4 - 5 + 6 ..., or
S_even - S_odd = -1 * (1 - 2 + 3 - 4 + 5 - 6 + ....)

but the expression in brackets is familiar: it is S2!

S_even - S_odd = -S2, or
S_even - S_odd = -1/4

Great, a system of 2 linear equations with 2 unknowns - easily solvable:

S_even + S_odd = -1/12
S_even - S_odd = -1/4

Adding them up:

2S_even = -1/12 - 1/4
2S_even = (-1-3)/12
2S_even = -1/3

(Beating drums..........)

S_even = -1/6

The only thing that this proves is the following: "Mathematics is imperfect. Mathematics is just another tool created by human beings to try to make sense of nature, but nature is way more advanced than human beings can ever comprehend and/or understand".

With Respect,
Marcelo