Low basis theorem - Revision history

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	~~In the mathematics of [[convergent series\|convergent]] and [[divergent series]], '''Euler summation''~~' is ~~a summability 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 Σ'~~'a~~''<sub>''n''</sub>, if its [[Euler transform]] converges to a sum, then that sum~~ is ~~called the '''Euler sum''' of the original series. As well as being used~~ to ~~define values for divergent series, Euler summation can be used to speed the convergence of series.~~

	~~Euler summation can be generalized into a family of methods denoted (E,~~ '~~'q''), where ''q'' ≥ 0. The (E, 0) sum is the usual (convergent) sum, while (E, 1) is~~ the ~~ordinary Euler sum~~. ~~All of these methods are strictly weaker than [[Borel summation]];~~ for ~~''q'' > 0 they are incomparable with [[Abel summation]]~~.

	~~==Definition==~~
	~~Euler summation is particularly used to [[series acceleration\|accelerate the convergence]] of alternating series and allows evaluating divergent sums.~~
	:<~~math~~> ~~_{E_y}\, \sum_{j=0}^\infty a_j := \sum_{i=0}^\infty \frac{1}{(1+y)^{i+1}} \sum_{j=0}^i {i \choose j} y^{j+1} a_j .~~<~~/math~~>

	~~To justify the approach notice that for interchanged sum, Euler's summation reduces to the initial series, because~~
	~~:<math>y^{j+1}\sum_{i=j}^\infty {i \choose j} \frac{1}{(1+y)^{i+1}}=1.</math>~~

	~~This method itself cannot be improved~~ by ~~iterated application, as~~
	~~:<math> _{E_{y_1}} {}_{E_{y_2}}\sum = \, _{E_{\frac{y_1 y_2}{1+y_1+y_2}}} \sum~~.~~</math>~~

	~~==Examples==~~

	* We have <math>\sum_{j=0}^\infty x^j P_k(j)= \sum_{i=0}^k \frac{x^i}{(1-x)^{i+1}}\sum_{j=0}^i {i \choose j} (-1)^{i-j} P_k(j) </math>, if <math>P_k</math> is a polynomial of [[Degree of a polynomial\|degree]] k. ~~Note that in this case Euler summation reduces an infinite series to a finite sum~~.

	* The particular choice <math>P_k(j):~~= (j+1)^k<~~/~~math> provides an explicit representation of the [[Bernoulli numbers]], since <math>\zeta(-k)= -\frac{B_{k+1}}{k+1}<~~/~~math>~~. ~~Indeed, applying Euler summation to the zeta function yields <math>\frac{1}{1-2^{k+1}}\sum_{i=0}^k \frac{1}{2^{i+1}} \sum_{j=0}^i {i \choose j} (-1)^j (j+1)^k <~~/~~math>, which is polynomial for <math>k<~~/~~math> a positive integer; cf. [[Riemann zeta function#Representations\|Riemann zeta function]]~~.

	* <math>\sum_{j=~~0}^\infty z^j= \sum_{i=0}^\infty \frac{1}{(1+y)^{i+1}} \sum_{j=0}^i {i \choose j} y^{j+1} z^j = \frac{y}{1+y} \sum_{i=0} \left( \frac{1+yz}{1+y} \right)^i<~~/~~math>. With an appropriate choice of <math>y</math> this series converges to <math>\frac{1}{1~~-~~z}<~~/~~math>.~~

	~~==See also==~~
	* [[Borel summation]]
	* [[Cesàro summation]]
	* [[Lambert summation]]
	* [[Perron's formula]]
	* [[Abelian and tauberian theorems]]
	* [[Abel–Plana formula]]
	* [[Abel's summation formula]]
	* [[Van Wijngaarden transformation]]

	~~==References==~~
	~~<div class="references-small">~~
	*{{cite book \|last=Korevaar \|first=Jacob \|title=Tauberian Theory: A Century of Developments \|publisher=Springer \|year=2004 \|isbn=3-540-21058-X}}
	*{{cite book \|author=Shawyer, Bruce and Bruce Watson \|title=Borel's Methods of Summability: Theory and Applications \|publisher=Oxford UP \|year=1994 \|isbn=0-19-853585-6}}
	*{{cite book \|author=Apostol, Tom M. \|title= Mathematical Analysis Second Edition \|publisher= Addison Wesley Longman \|year=1974 \|isbn=0-201-00288-4}}

	</~~div>~~

	~~[[Category:Mathematical series]]~~
	~~[[Category:Summability methods]~~]

en>CBM: This article uses Harvard-style references

2013-04-14T18:37:56Z

This article uses Harvard-style references

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+{{more footnotes|date=September 2012}}
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+Euler summation can be generalized into a family of methods denoted (E, ''q''), where ''q'' ≥ 0. The (E, 0) sum is the usual (convergent) sum, while (E, 1) is the ordinary Euler sum. All of these methods are strictly weaker than [[Borel summation]]; for ''q'' > 0 they are incomparable with [[Abel summation]].
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+==Definition==

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