Sum of all natural numbers is 0?

Question

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.

Answer 1

If you assume the sum of all naturals is finite and the "usual arithmetic" rules here, you can get pretty awesome results, like:

$$S=1+2+3+\ldots\implies 2+4+6+\ldots=2(1+2+3+\ldots)=2S\implies$$

$$S=1+2+3+\ldots=1+3+5+\ldots+2+4+6+\ldots=1+3+5+\ldots+2S\implies$$

$$-S=1+3+5+\ldots$$

And a lot more of whatever else you want, say:

$$1+1+1+\ldots-S=1+1+1+\ldots+1+3+5+\ldots=1+1+3+1+5+1+\ldots=$$

$$=2+4+6+\ldots=2S\implies 3S=1+1+1+\ldots\;,\;\;\text{etc.} $$

Answer 2

The most common way to define $\sum_{n\geq1}n$ is by its partial sums: $$ \sum_{n\geq1}n\equiv\lim_{N\rightarrow\infty}s_{N}\text{ where }s_{N}\equiv\sum_{n=1}^{N}n=\frac{1}{2}N\left(N+1\right). $$ Clearly, this is divergent.

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As you seem to have noticed, there are other ways to define this sum. This causes all sorts of confusion, since people like to also use the notation $\sum_{n\geq1}n$ in these alternative definitions.

Let's now look at "zeta function regularization", which is the interpretation which yields $-1/12$. In particular, note that $$ \zeta(s)=\sum_{n\geq1}n^{-s}\text{ for }\operatorname{Re}(s)>1 $$ by definition. In the above, we are using the usual notion of summation by a limit of partial sums. The $\zeta$ function, however, is also defined for other values of $s$ by analytic continuation. We can thus reinterpret $$ \check{\sum_{n\geq1}}n\equiv\zeta(-1)=-1/12. $$ However, note that the two interpretations $$\sum\text{ and }\check{\sum}$$ are completely different!

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As for the method which you refer to as "abusing divergent and oscillating series", perhaps you are thinking of this. This is not rigorous.

Answer 3

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.

Answer 4

The error in your derivation - I kid you not - is the assertion that 1+2=3 (and similarly 2+3=5, 3+4=7 etc.)

Sums like this get their meaning from Zeta function regularization, and the intuitive algebraic manipulations are only shorthand for correct manipulations on corresponding Dirichlet series. It so happens that the manipulations in the linked proof can be translated to proper ones, while your manipulations cannot.

In particular, it's easier to work with alternating sums because they can be thought of as formal power series in which you then substitute 1 (and since shifting is equivalent to multiplying by a power of $x$, you can do that freely). But you can't do that for sums like 1+2+3+... because you would get a 0 denominator, so other techniques are needed.

If your manipulations are translated to Zeta Function regularization, you'll get that the 1 and 2 you try to add are not really 1 and 2, they are $1^{-s}$ and $2^{-s}$, and these do not add up to $3^{-s}$, rendering the rest of the derivation invalid.

See also https://en.wikipedia.org/wiki/1_+2+3+4+_%E2%8B%AF#Heuristics.

Answer 5

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...)

Answer 6

In some sense, the error is in the subtraction. You're trying to do some arithmetic on things which are no longer proper numbers, and not everything behaves the same.

For a mild simplification, think about the sum $$ S = 1 + 1 + 1 + \cdots $$ Then by the same reasoning as yours, we can consider $$ 2S := S + S = 1 + 1 + 1 + \cdots $$ Here I've interleaved sums, and this is okay because all terms are positive, though not valid in general. This addition of $S$ with itself in some sense is fine and we see $2S = S$. Then you'd like to say that $$ 2S = S \implies S = 0, $$ but the problem it doesn't for arbitrary "quantities" S, including the case that $S = +\infty$, which is the value of $S$ here. (The above statement doesn't even hold in many finite algebraic systems like $\mathbb Z/6 \mathbb Z$.)

Weird things happen when you try to work with infinities, and not everything commutes. In particular, you need to be careful about subtracting infinite sums because conditionally convergent sums can be rearranged to take arbitrary values.

Answer 7

Note that 4S = S may also mean both are Infinite as

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

Answer 8

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.