We have this sequence:
S1: 1+2+3+4+5+6.. (to infinity)
It has been demonstrated, that S1 = -1/12.
Now, what happens if i multiply by a factor of 2?
S2: 2+4+6+8+10+12.... (to infinity).
I have 2S1, which is equal to -1/6
On this, we can create a equation for the odd numbers:
S3: 1+3+5+5+7+9+11... (to infinity)
We know that for every term in S2, every term in S3 is just (n-1)
Or, The sum of the even numbers, Minus , the sum of infinitely many (-1)s
So S3 = -1/6 - ∞
However, we also know that the odd numbers + the even numbers = The natural numbers.
So let's try it.
-1/6 - (-1/6 -∞)
We have -1/6 + 1/6 + ∞
Which is just ∞
So, there we have it. a paradox. S1 cannot be both -1/12 or ∞