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In 1890, Ernesto Cesàro stated a broader family of summation methods which have since been called (C, α) for non-negative integers α. The (C, 0) method is just ordinary summation, and (C, 1) is Cesàro summation as described above. The higher-order methods can be described as follows: given a series Σa n, define the quantities
For any summation method L, its Abelian theorem is the result that if c = (c n) is a convergent sequence, with limit C, then L(c) = C. [clarification needed]An example is given by the Cesàro method, in which L is defined as the limit of the arithmetic means of the first N terms of c, as N tends to infinity.
Euler–Boole summation is a method for summing alternating series. The concept is named after Leonhard Euler and George Boole. Boole published this summation method, using Euler's polynomials, but the method itself was likely already known to Euler. [1] [2] Euler's polynomials are defined by [1]
In mathematical analysis, a summability method is an alternative formulation of convergence of a series which is divergent in the conventional sense. Subcategories This category has the following 2 subcategories, out of 2 total.
A generalized definition of the "sum" of a divergent series is called a summation method or summability method. There are many different methods and it is desirable that they share some properties of ordinary summation.
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators.
In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σa n, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original ...
More generally one can define summation methods slightly stronger than Borel's by taking the numbers b n to be slightly larger, for example b n = cnlog n or b n =cnlog n log log n. In practice this generalization is of little use, as there are almost no natural examples of series summable by this method that cannot also be summed by Borel's method.