Sum of all natural numbers is 0? A fairly well-known (and perplexing) fact is that the sum of all natural numbers is given the value -1/12, at least in certain contexts. Quite often, a "proof" is given which involves abusing divergent and oscillating series, and the numbers seem to fit at the end. (There are also other proofs involving the zeta function or other things, but that's not my concern right now).
However, I tried to calculate it my own way using similar methods, and this is what I got:
$$\begin{align}
S = 1 &+ 2 + 3 + 4 + …\\\\
2S = 1 &+ 2 + 3 + 4 + 5 + \ldots\\
       &+ 1 + 2 + 3 + 4 + \ldots\\\\
   = 1 &+ 3 + 5 + 7 + 9 + \ldots\\
\end{align}$$
Also,
$2S = 2 + 4 + 6 + 8 + 10 + \ldots$
Adding the above together,
$$4S = 1 + 2 + 3 + 4 + 5 + … = S$$
Which means $3S = 0$, therefore $S = 0$
Obviously, there must be some reason why this is not ok, otherwise we'd have $-1/12 = 0$.
But why is my method wrong while the one involving oscillating series is considered acceptable?
Additional clarification: I was wondering if there are specific ways to manipulate this kind of series such that by assuming that the result is finite and performing only certain types of "permitted" operations, one could be confident to get to the same result as the rigorous way of assigning a finite value to the sum of the series. So far, the consensus seems to be negative.

The simple answer to what seems to be your question: "why is my method wrong?"
You are assuming that a clearly divergent series converges to some number $S$; this provides a contradiction, if you assume a contradiction to be true then you can essentially use it to prove whatever nonsense you want.
Showing that this series sums to $-1/12$ and using this result does not have much to do with what you are thinking of as the sum of an infinite series. The Numberphile video is entertaining, but it is trash as far as actual mathematics. They don't explain this result, the meaning behind it, the context in which it is used, or the justification for the steps they use in their "proof".  There is nothing rigorous about the proof they present, it's all smoke and mirrors. So it is natural that, by imitating their proof, you are ending up with a wrong result.
