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Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.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, for which conventional summation is undefined.
Ramanujan summation is a method to isolate the constant term in the Euler–Maclaurin formula for the partial sums of a series. For a function f , the classical Ramanujan sum of the series ∑ k = 1 ∞ f ( k ) {\displaystyle \textstyle \sum _{k=1}^{\infty }f(k)} is defined as
In number theory, Ramanujan's sum, usually denoted c q (n), is a function of two positive integer variables q and n defined by the formula = (,) =, where (a, q) = 1 means that a only takes on values coprime to q.
In the example, there are 3 summation indices , and because the integrand is a product of 3 series expansions. [16] The free summation indices (variables) are the summation indices that remain after completing all integrations. Integration reduces the number of sums in the integrand by replacing the series expansions (sums) with an integration ...
Srinivasa Ramanujan Aiyangar [a] (22 December 1887 – 26 April 1920) was an Indian mathematician.Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then ...
Download as PDF; Printable version; ... Ramanujan summation; Regularization (physics) ... Text is available under the Creative Commons Attribution-ShareAlike 4.0 ...
However, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. For example, the Ramanujan summation of this series is −1, which is the limit of the series using the 2-adic metric.
There are several different summation methods called zeta function regularization for defining the sum of a possibly divergent series a 1 + a 2 + ..... One method is to define its zeta regularized sum to be ζ A (−1) if this is defined, where the zeta function is defined for large Re(s) by