Interpretation of Ramanujan summation of infinite divergent series

Question

I am not mathematician by any means so this question might be rather stupid. I came across this Wikipedia article on Ramanujan's summation and found this bewildering formula,

$$1 + 2 + 3 + \dots = - \frac1{12}$$

The article also says that "Ramanujan summation of a divergent series is not a sum in the traditional sense". I am wondering then what this summation actually implies?

Is there any interpretation of this confounding result in the physical world that someone not so math-savvy can understand?

Answer 1

This result may be counterintuitive and makes sense at a high level of complex analysis dealing with Zeta function's and analytic number theory. Btw this has not only to do with math, this is even used in Physic but again at a very high level when we treat string theory or the Casimir effect (and other phenomena that can be solved using Zeta regularization).

If you're asking yourself why those kind of results make sense that's why in math we usually like to generalize things: when we have something that doesn't make sense in the "usual way" we give it a new sense with the only condition that the other results are preserved. In this case Ramanujan summation (along with other formulas like Cesaro or Abel summation) gives the usual results with convergent series and new results with divergent ones, that can be used in the same way as the others.

Answer 2

There are answers in -as I think- in mathoverflow for "what are physical interpretations of zeta at negative arguments" (or the like, I don't have the actual words in mind). Just browse a bit through MO or MSE using that terms as filter.

---

0) Need of a new concept - and a basic requirement: Make the general term explicite

For the interpretation and assignment of a meaningful value in your series you must define, how the general term, say $a_k$, shall be constructed, i.e. how the value of $a_k$ is dependend on its index $k$. This is a requirement which shall be demanded in all discussions of divergent series (and should not been forgot when such series, written only in their "obvious" numerical forms, are considered at all).

In this example the value of the general term, say $a_k$, reduces simply to the index $k$ if taken from $k=1 \ldots \infty $ and we have actually the formal statement $S = \sum_{k=1}^\infty k$. But because the summing procedure is obviously divergent we must introduce some new concept if we want assign something meaningful to it.

1) Geometric series

A useful idea here was (for instance already by L. Euler) that we can cofactor a variable $x$ at the terms $a_k$ such that for some $x$ the series becomes convergent and evaluatable to some finite value and see what happens with the sequence of resulting sum-values, when the expressions of $x$ approximate $1$ . For instance, we could redefine your sum as $$ S(x) = 1 + 2x + 3x^2 + 4x^3 + ... \\ S = \lim_{x \to 1^-} S(x) $$ because there are some $x$ (actually it is important, that there exists a continuous interval of $x$) where this is convergent, for instance for $x=0 \ldots \frac12$. We can then even observe that with that definition the sum $S(x)$ has a closed form $$S(x) = {d\over dx}{1\over 1-x } = {1\over (1-x)^2 }$$ This is already a very nice representation, because it allows now even negative $x$ which result in finite values for the alternating sum, even if divergent, because it is accepted to assign that fraction's value also to its formal power series-expression even if the latter is divergent - except, well, except if $x=1$.
But what we want here is actually just the latter, so we have not yet a satisfying answer.

2) Dirichlet series

Another idea is to apply a function of $x$ to the exponent of the terms $a_k$ and then let $x$ go to $1$ . Thus we consider the notation $$ S(x) = 1^x + 2^x + 3^x +4^x + ... $$ which is convergent for a continuous interval of $x$, for instance of $x = -\infty \ldots -2$ (and even a bit more). But still, for $x=1$ we get no obvious answer.
But interestingly, for the whole interval of convergence we have also the functional relation $$ \begin{matrix} \text{let } &A(x) &=& 1^x - 2^x + 3^x - 4^x + \ldots \\ \text{then } &S(x) &=& A(x) + 2 \cdot 2^x \cdot S(x) \\ \text{and} & S(x) &=& A(x)/(1-2 \cdot 2^x) \end{matrix} $$ And now we can assign a meaningful value to $S = \lim_{x \to 1^- }$. Either by evaluating $A(1)$ as conditionally converging series in the given form, or by the above geometric-series interpretation and its derivative at $x=-1$ where both ways of evaluation give the same rational expression $A(1)=\frac 14$.
After that nothing more divergent is there and we get $$ S = \lim_{x \to 1^-} S(x) = \frac 14 / (1-2\cdot 2^x) = \frac 14 / (1-4)= - \frac 1{12} $$ having now a proposal for a meaningful assignment of a finite value for the infinite divergent series.

---

3) Caveat

Of course that value must conform with all and any cases where such series occur in mathematics, and one example of a contradiction with known results taken from the mathematic without divergent summation would invalidate that procedure!
Moreover, we would hope that even in the physical world, where we model some observations with that series, such an assignement of values to a divergent series would hold. Interestingly such observations exist in the real world and it seems that the whole process and also the final value meets the modeling of that observations. (Examples are given to the according questions either in MSE here or in mathoverflow (I've just found (1)(2)), you can do a search for the important words "zeta" and "at negative arguments" or similar)