sequences and series - The sum of all the odd numbers to infinity

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 ∞