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


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