Ramanujan summation is a technique invented by the mathematician
Srinivasa Ramanujan for assigning a value to
divergent infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
, for which conventional summation is undefined.
Summation
Since there are no properties of an entire sum, the Ramanujan summation functions as a property of partial sums. If we take the
Euler–Maclaurin summation formula together with the correction rule using
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s, we see that:
:
Ramanujan wrote it for the case ''p'' going to infinity:
:
where ''C'' is a constant specific to the series and its analytic continuation and the limits on the integral were not specified by Ramanujan, but presumably they were as given above. Comparing both formulae and assuming that ''R'' tends to 0 as ''x'' tends to infinity, we see that, in a general case, for functions ''f''(''x'') with no divergence at ''x'' = 0:
:
where Ramanujan assumed
By taking
we normally recover the usual summation for convergent series. For functions ''f''(''x'') with no divergence at ''x'' = 1, we obtain:
:
''C''(0) was then proposed to use as the sum of the divergent sequence. It is like a bridge between summation and integration.
The convergent version of summation for functions with appropriate growth condition is then:
:
Sum of divergent series
In the following text,
indicates "Ramanujan summation". This formula originally appeared in one of Ramanujan's notebooks, without any notation to indicate that it exemplified a novel method of summation.
For example, the
of is:
:
Ramanujan had calculated "sums" of known divergent series. It is important to mention that the Ramanujan sums are not the sums of the series in the usual sense,
i.e. the partial sums do not converge to this value, which is denoted by the symbol
In particular, the
sum of was calculated as:
:
Extending to positive even powers, this gave:
:
and for odd powers the approach suggested a relation with the
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s:
:
It has been proposed to use of ''C''(1) rather than ''C''(0) as the result of Ramanujan's summation, since then it can be assured that one series
admits one and only one Ramanujan's summation, defined as the value in 1 of the only solution of the difference equation
that verifies the condition
.
[ Éric Delabaere]
Ramanujan's Summation
''Algorithms Seminar 2001–2002'', F. Chyzak (ed.), INRIA, (2003), pp. 83–88.
This demonstration of Ramanujan's summation (denoted as
) does not coincide with the earlier defined Ramanujan's summation, ''C''(0), nor with the summation of convergent series, but it has interesting properties, such as: If ''R''(''x'') tends to a finite limit when ''x'' → 1, then the series
is convergent, and we have
: