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.
I think such issues, which superficially seem to be about symbol-manipulation, are more genuinely about operations within a context in which "infinite sums" (for example) are characterized in various ways. That is, to say $\sum\_n a\_n=A$ should mean *what*? It is certainly true that the "populist" answer is convergence-of-sequence-of-finite-subsums, where convergence of a sequence is fairly intuitive (whatever its formal "definition/characterization" may be).

Thus, in one context (or more), the infinite sum has no sense as a real or complex number. Ok.

In other senses, going back at least to Euler, and resurrected by Ramanujan post-Weierstrass, when one a-priori acknowledges the non-convergence (=lack of useful, manipulable sense, in terms of many *characterizations*...), then one might aggressively ask about other *interpretations* (... contexts...)

So Euler already knew that, in some very particular sense, the values of zeta(s) at negative integers are what we know understand them to be... by analytic continuation, although the latter notion did not exist in Euler's time (so far as I know).

The genuine operational point is to maybe not just play games with symbols so much, but, rather, to try to say what real thing one's symbols encapsulate, so that the symbol-motion is just book-keeping about a real thing, rather than a fiction with nothing underlying.

To say that manipulation of "non-convergent sums" is not legitimate is too glib, since it assumes a certain interpretation which is in many regards the strictest possible. Much like declaring as much stuff as possible "nonsense" just to avoid having to grapple with it.

Still, again, to just shuffle symbols is not so interesting... not at all because of possible "lack of rules [sic]", but because it might be without context, and/or without characterization of the phenomena purportedly described by the symbols.

(My colleagues who describe various ideas as "just marks on the page" dismay me...)