Question: A fairly well-known (and perplexing) fact is that [the sum of all natural numbers is given the value -1/12](https://math.stackexchange.com/q/39802/31408), at least in certain contexts. Quite often, a ["proof"](https://www.youtube.com/watch?v=w-I6XTVZXww) 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.

Answer: 
A whole book could be written about it. In fact, it has:

G.H Hardy, *Divergent Series*, (Oxford, 1949).

Euler (to name but one) was far less uptight about assigning values to divergent series (such as $1+2+4+8+16…=-1$) than today's mimsy pedants. Cauchy, Abel and others took the subject further and with more rigour.

But I am not going to summarize the entire book here.
Note that 4S = S may also mean both are Infinite as 

```
N*infinite = Infinite ; N being a +Ve Real Value  

```