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| {{more footnotes|date=September 2012}}
| | The author's title is Christy. Some time in the past she chose to reside in Alaska and her mothers and fathers live close by. It's not a typical thing but what I like performing is to climb but I don't have the time lately. Distributing production has been his occupation for some time.<br><br>Stop by my homepage ... [http://afeen.fbho.net/v2/index.php?do=/profile-210/info/ real psychic] |
| 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.
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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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| Euler summation is particularly used to [[series acceleration|accelerate the convergence]] of alternating series and allows evaluating divergent sums.
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| :<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>
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| To justify the approach notice that for interchanged sum, Euler's summation reduces to the initial series, because
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| :<math>y^{j+1}\sum_{i=j}^\infty {i \choose j} \frac{1}{(1+y)^{i+1}}=1.</math>
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| This method itself cannot be improved by iterated application, as
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| :<math> _{E_{y_1}} {}_{E_{y_2}}\sum = \, _{E_{\frac{y_1 y_2}{1+y_1+y_2}}} \sum.</math>
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| ==Examples==
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| * 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.
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| * 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]].
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| * <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>.
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| ==See also==
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| * [[Borel summation]]
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| * [[Cesàro summation]]
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| * [[Lambert summation]]
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| * [[Perron's formula]]
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| * [[Abelian and tauberian theorems]]
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| * [[Abel–Plana formula]]
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| * [[Abel's summation formula]]
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| * [[Van Wijngaarden transformation]]
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| ==References==
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| <div class="references-small">
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| *{{cite book |last=Korevaar |first=Jacob |title=Tauberian Theory: A Century of Developments |publisher=Springer |year=2004 |isbn=3-540-21058-X}}
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| *{{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}}
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| *{{cite book |author=Apostol, Tom M. |title= Mathematical Analysis Second Edition |publisher= Addison Wesley Longman |year=1974 |isbn=0-201-00288-4}}
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| </div>
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| [[Category:Mathematical series]]
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| [[Category:Summability methods]]
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The author's title is Christy. Some time in the past she chose to reside in Alaska and her mothers and fathers live close by. It's not a typical thing but what I like performing is to climb but I don't have the time lately. Distributing production has been his occupation for some time.
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