Ramanujan summation

Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a sum to infinite divergent series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.

Ramanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as it doesn't exist. If we take the Euler–Maclaurin summation formula together with the correction rule using Bernoulli numbers, we see that:

:$frac\{fleft(\; 0\; ight)\; \}\{2\}+fleft(\; 1\; ight)\; +cdots+fleft(\; n-1\; ight)\; +frac\{fleft(\; n\; ight)\; \}\{2\}\; =frac\{fleft(\; 0\; ight)\; +fleft(\; n\; ight)\; \}\{2\}+sum\_\{k=1\}^\{n-1\}fleft(k\; ight)\; =\; int\_0^n\; f(x),dx\; +\; sum\_\{k=1\}^pfrac\{B\_\{k+1\{(k+1)!\}left(f^\{(k)\}(n)-f^\{(k)\}(0)\; ight)+R$

or simply::$fleft(\; 1\; ight)+fleft(\; 2\; ight)\; +cdots+fleft(\; n-2\; ight)\; +\; fleft(\; n-1\; ight)=\; C\; +\; int\_1^n\; f(x),dx\; +\; sum\_\{k=1\}^inftyfrac\{B\_\{k+1\{(k+1)!\}left(f^\{(k)\}(n)-f^\{(k)\}(0)\; ight)$

Where C is a constant specific to the series and its analytic continuum. This he proposes to use as the sum of the divergent sequence. It is like a bridge between summation and integration. Using standard extensions for known divergent series, he calculated "Ramanujan summation" of those. In particular, the sum of 1 + 2 + 3 + 4 + · · · is

:$1+2+3+cdots\; =\; -frac\{1\}\{12\}\; (Re)$

where the notation $(Re)$ indicates Ramanujan summation. [ Éric Delabaere, [http://algo.inria.fr/seminars/sem01-02/delabaere2.pdf Ramanujan's Summation] , "Algorithms Seminar 2001–2002", F. Chyzak (ed.), INRIA, (2003), pp. 83–88.] This formula originally appeared in one of Ramanujan's notebooks, without any notation to indicate that it was a Ramanujan summation.

For even powers we have:

:$1+2^\{2k\}+3^\{2k\}+cdots\; =\; 0\; (Re)$

and for odd powers we have a relation with the Bernoulli numbers:

:$1+2^\{2k-1\}+3^\{2k-1\}+cdots\; =\; -frac\{B\_\{2k\{2k\}\; (Re).$

See also

* Borel summation
* Cesàro summation
* Ramanujan's sum

References