Originally shared by Terence Tao
There is a lot of discussion in various online mathematical forums currently about the interpretation, derivation, and significance of Ramanujan’s famous (but extremely unintuitive) formula
1+2+3+4+… = -1/12 (1)
or similar divergent series formulae such as
1-1+1-1+… = 1/2 (2)
or
1+2+4+8+… = -1. (3)
One can view this topic from either a pre-rigorous, rigorous, or post-rigorous perspective (see this page of mine for a description of these three terms: http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ ). The pre-rigorous approach is not particularly satisfactory: here one is taught the basic rules for manipulating finite sums (e.g. how to add or subtract one finite sum from another), and one is permitted to blindly apply these rules to infinite sums. This approach can give derivations of identities such as (1), but can also lead to derivations of even more blatant absurdities such as 0=1, which of course makes any similar derivation of (1) look quite suspicious.
From
a rigorous perspective, one learns in undergraduate analysis classes the notion of a convergent series and a divergent series, with the former having a well defined limit, which enjoys most of the same laws of series that finite series do (particularly if one restricts attention to absolutely convergent series). In more advanced courses, one can then learn of more exotic summation methods (e.g. Cesaro summation, p-adic summation or Ramanujan summation) which can sometimes (but not always) be applied to certain divergent series, and which obey some (but not all) of the rules that finite series or absolutely convergent series do. One can then carefully derive, manipulate, and use identities such as (1), so long as it is made precise at any given time what notion of summation is in force. For instance, (1) is not true if summation is interpreted in the classical sense of convergent series, but it is true for some other notions of convergence, such as Ramanujan convergence, or a real-variable analogue of that convergence that I describe in this post:
From a post-rigorous perspective, I believe that an equation such as (1) should more accurately be rendered as
1+2+3+4+… = -1/12 + …
where the “…” on the right-hand side denotes terms which could be infinitely large (or divergent) when interpreted classically, but which one wishes to view as “negligible” for one’s intended application (or at least “orthogonal” to that application). For instance, as a rough first approximation (and assuming implicitly that the summation index in these series starts from n=1 rather than n=0), (1), (2), (3) should actually be written as
1+2+3+4+… = -1/12 + 1/2 infinity^2 (1)’
1-1+1-1+… = 1/2 – (-1)^{infinity} /2 (2)’
or
1+2+4+8+… = -1 + 2^{infinity} (3)’
and more generally
1+x+x^2+x^3+… = 1/(1-x) + x^{infinity}/(x-1)
where the terms involving infinity do not make particularly rigorous sense, but would be considered orthogonal to the application at hand (a physicist would call these quantities unphysical) and so can often be neglected in one’s manipulations. (If one wanted to be even more accurate here, the 1/2 infinity^2 term should really be the integral of x dx from 0 to infinity.) To rigorously formalise the notion of ignoring certain types of infinite expressions, one needs to use one of the summation methods mentioned above (with different summation methods corresponding to different classes of infinite terms that one is permitted to delete); but the above post-rigorous formulae can still provide clarifying intuition, once one has understood their rigorous counterparts. For instance, the formulae (1)’ and (3)’ are now consistent with the left-hand side being positive and diverging to infinity, and the formula (2)’ is consistent with the left-hand side being indeterminate in limit, with both 0 and 1 as limit points. The fact that divergent series often do not behave well with respect to shifting the series can now be traced back to the fact that the infinite terms in the above identities produce some finite remainders when the infinity in those terms is shifted, say to infinity+1.
For a more advanced example, I believe that the “field of one element” should really be called “the field of 1+… elements”, where the … denotes an expression which one believes to be orthogonal to one’s application.