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A summation method is any method for assigning sums to divergent series in a way that systematically extends the classical notion of the sum of a series. Summation methods include Cesàro summation, generalized Cesàro (,) summation, Abel summation, and Borel summation, in order of applicability to increasingly divergent series.
In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
In mathematical analysis, Cesàro summation (also known as the Cesàro mean [1] [2] or Cesàro limit [3]) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.
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.
Summation by parts is frequently used to prove Abel's theorem and Dirichlet's test. One can also use this technique to prove Abel's test: If is a convergent series, and a bounded monotone sequence, then = = converges. Proof of Abel's test.
The geometric series is an infinite series derived from a special type of sequence called a geometric progression.This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term , and the next one being the initial term multiplied by a constant number known as the common ratio .
Borel summation requires that the coefficients do not grow too fast: more precisely, a n has to be bounded by n!C n+1 for some C. There is a variation of Borel summation that replaces factorials n! with (kn)! for some positive integer k, which allows the summation of some series with a n bounded by (kn)!C n+1 for some C.