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

Wikimedia Foundation. 2010.

Look at other dictionaries:

Ramanujan's sum — This article is not about Ramanujan summation. In number theory, a branch of mathematics, Ramanujan s sum, usually denoted c q ( n ), is a function of two positive integer variables q and n defined by the formula:c q(n)=sum {a=1atop… … Wikipedia
Srinivasa Ramanujan — Infobox Scientist name=Srinivasa Ramanujan thumb|Srinivasa Ramanujan birth date = birth date|1887|12|22|df=y birth place = Erode, Tamil Nadu, India death date = death date and age|1920|4|26|1887|12|22|df=y death place = Chetput, (Madras), Tamil… … Wikipedia
List of topics named after Srinivasa Ramanujan — Srinivasa Ramanujan (1887 1920) is the eponym of all of the topics listed below.*Dougall Ramanujan identity *Hardy Ramanujan number *Landau Ramanujan constant *Ramanujan s congruences *Ramanujan Nagell equation *Ramanujan Peterssen conjecture… … Wikipedia
Sumatorio de Ramanujan — Srinivasa Ramanujan. El sumatorio de Ramanujan es una técnica inventada por el matemático indio Srinivasa Ramanujan para asignar una suma a una serie divergente infinita. A pesar de que el sumatorio de Ramanujan de una serie divergente no es una… … Wikipedia Español
List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… … Wikipedia
List of Indian inventions — [ thumb|200px|right|A hand propelled wheel cart, Indus Valley Civilization (3000–1500 BCE). Housed at the National Museum, New Delhi.] [ 200px|thumb|Explanation of the sine rule in Yuktibhasa .] List of Indian inventions details significant… … Wikipedia
Arithmetic function — In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers (i.e. positive integers) that expresses some arithmetical property of n. [1] An example of an arithmetic… … Wikipedia
Polylogarithm — Not to be confused with polylogarithmic. In mathematics, the polylogarithm (also known as Jonquière s function) is a special function Lis(z) that is defined by the infinite sum, or power series: It is in general not an elementary function, unlike … Wikipedia
Ramanujansumme — Als Ramanujansumme wird in der Zahlentheorie, einem Teilgebiet der Mathematik, eine bestimmte endliche Summe cq(n), deren Wert von der natürlichen Zahl q und der ganzen Zahl n abhängt, bezeichnet. Sie wird durch definiert. Die Schreibweise (a,q)… … Deutsch Wikipedia
Riemann hypothesis — The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011 … Wikipedia

Wikipedia

Ramanujan summation

Look at other dictionaries:

Share the article and excerpts

Direct link