Summability methods dFrom Wikipedia, the free encyclopediaContents1 Darbouxs formula 11.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Dimensional regularization 22.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Divergent series 43.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Theorems on methods for summing divergent series . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Properties of summation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Classical summation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4.1 Absolute convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4.2 Sum of a series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5 Nrlund means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5.1 Cesro summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.6 Abelian means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.6.1 Abel summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6.2 Lindelf summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.7 Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.7.1 Analytic continuation of power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.7.2 Euler summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.7.3 Analytic continuation of Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.7.4 Zeta function regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.8 Integral function means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.8.1 Borel summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.8.2 Valirons method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.9 Moment methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.9.1 Borel summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.10Miscellaneous methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10iii CONTENTS3.10.1 Hausdor transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.10.2 Hlder summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.10.3 Huttons method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.10.4 Ingham summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.10.5 Lambert summability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.10.6 Le Roy summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.10.7 Mittag-Leer summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.10.8 Ramanujan summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.10.9 Riemann summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.10.10 Riesz means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.10.11 Valle-Poussin summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.14Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 143.14.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.14.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.14.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Chapter 1Darbouxs formulaNot to be confused with ChristoelDarboux formula.In mathematical analysis, Darbouxsformula is a formula introduced by Gaston Darboux (1876) for summinginnite series by using integrals or evaluating integrals using innite series. It is a generalization to the complex planeof the EulerMaclaurin summation formula, which is used for similar purposes and derived in a similar manner (byrepeated integration by parts of a particular choice of integrand). Darbouxs formula can also be used to derive theTaylor series of the calculus.1.1 StatementIf (t) is a polynomial of degree n and f an analytic function thennm=0(1)m(z a)m[(nm)(1)f(m)(z) (nm)(0)f(m)(a)]=(1)n(z a)n+110(t)f(n+1)[a + t(z a)]dt.The formula can be proved by repeated integration by parts.1.2 Special casesTaking to be a Bernoulli polynomial in Darbouxs formula gives the EulerMaclaurin summation formula. Taking to be (t 1)ngives the formula for a Taylor series.1.3 ReferencesDarboux (1876), Sur les dveloppements en srie des fonctions d'une seule variable, Journal de Mathma-tiques Pures et Appliques 3 (II): 291312Whittaker, E. T. and Watson, G. N. A Formula Due to Darboux. 7.1 in A Course in Modern Analysis, 4thed. Cambridge, England: Cambridge University Press, p. 125, 1990.1.4 External linksDarbouxs formula at MathWorld1Chapter 2Dimensional regularizationIn theoretical physics, dimensional regularizationis a method introduced by Giambiagi and Bollini[1] for regularizingintegrals in the evaluation of Feynman diagrams; in other words, assigning values to them that are meromorphic func-tions of an auxiliary complex parameter d, called (somewhat confusingly) the dimension.Dimensional regularization writes a Feynman integral as an integral depending on the spacetime dimension d andthe squared distances (xixj)2of the spacetime points xi, ...appearing in it.In Euclidean space, the integral oftenconverges for Re(d) suciently large, and can be analytically continued from this region to a meromorphic functiondened for all complex d.In general, there will be a pole at the physical value (usually 4) of d, which needs to becanceled by renormalization to obtain physical quantities. Etingof (1999) showed that dimensional regularization ismathematically well dened, at least in the case of massive Euclidean elds, by using the BernsteinSato polynomialto carry out the analytic continuation.There is a tradition of confusing the parameter d appearing in dimensional regularization, which is a complex number,with the dimension of spacetime, which is a xed positive integer (such as 4). The reason is that if d happens to bea positive integer, then the formula for the dimensionally regularized integral happens to be correct for spacetimeof dimension d. For example, the surface area of a unit (d 1)-sphere is2d/2(d2)where is the gamma functionwhen d is a positive integer, so in dimensional regularization it is common to say that this is the surface area of asphere in d dimensions even when d is not an integer. Whereas there is no such thing as a sphere in non-integraldimensions, the formulas such as this are nonetheless a useful mnemonics in dimensional regularization. This failureto distinguish between the dimension of spacetime and the formal parameter d has led to speculation about spacetimesof non-integral dimension.If one wishes to evaluate a loop integral which is logarithmically divergent in four dimensions, likeddp(2)d1(p2+ m2)2,one rst rewrites the integral in some way so that the number of variables integrated over does not depend on d, andthen we formally vary the parameter d, to include non-integral values like d = 4 .This gives0dp(2)42(4)/2(42)p3(p2+ m2)2=2421sin(2)(1 2)m=182 1162(ln m24+ )+O().Emilio Elizalde has shown that Zeta regularization and dimensional regularization are equivalent since they use thesame principle of using analytic continuation in order for a series or integral to converge.2.1 Notes[1] Bollini 1972, p. 20.22.2. REFERENCES 32.2 ReferencesBollini, Carlos; Giambiagi, Juan Jose (1972), Dimensional Renormalization: The Number of Dimensions as aRegularizing Parameter. 12, Il Nuovo Cimento B, pp. 2026Etingof, Pavel (1999), Note on dimensional regularization, Quantum elds and strings: a course for math-ematicians, Vol. 1,(Princeton, NJ, 1996/1997), Providence, R.I.: Amer. Math. Soc., pp. 597607, ISBN978-0-8218-2012-4, MR 1701608Hooft, G. 't; Veltman, M. (1972), Regularization and renormalization of gauge elds, Nuclear Physics B 44(1): 189213, Bibcode:1972NuPhB..44..189T, doi:10.1016/0550-3213(72)90279-9, ISSN 0550-3213Kleinert, H.; Schulte-Frohlinde, V. (2001), Critical Properties of 4-Theories, pp. 1474, ISBN 978-981-02-4659-4, Paperpack ISBN 978-981-02-4659-4 (also available online). Read Chapter 8 for DimensionalRegularization.Chapter 3Divergent seriesFor the media franchise, see Divergent trilogy.For an elementary calculus-based introduction, see Divergent series on WikiversityLes sries divergentes sont en gnral quelque chose de bien fatal et cest une honte quon ose y fonder aucunedmonstration. (Divergent series are in general something fatal, and it is a disgrace to base any proof on them.Often translated as Divergent series are an invention of the devil ")N. H. Abel, letter to Holmboe, January 1826, reprinted in volume 2 of his collected papers.In mathematics, a divergent series is an innite series that is not convergent, meaning that the innite sequence ofthe partial sums of the series does not have a nite limit.If a series converges, the individual terms of the series must approach zero. Thus any series in which the individualterms do not approach zero diverges. However, convergence is a stronger condition: not all series whose termsapproach zero converge. A counterexample is the harmonic series1 +12+13+14+15+ =n=11n.The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme.In specialized mathematical contexts, values can be objectively assigned to certain series whose sequence of partialsums diverges, this is to make meaning of the divergence of the series. A summability method or summationmethod is a partial function from the set of series to values. For example, Cesro summation assigns Grandisdivergent series1 1 + 1 1 + the value1/2. Cesro summation is an averaging method, in that it relies on the arithmetic mean of the sequence ofpartial sums. Other methods involve analytic continuations of related series. In physics, there are a wide variety ofsummability methods; these are discussed in greater detail in the article on regularization.3.1 History... before Cauchy mathematicians asked not 'How shall we dene 11+1...?' but 'What is 11+1...?' and that thishabit of mind led them into unnecessary perplexities and controversies which were often really verbal.G. H. Hardy, Divergent series, page 643.2. THEOREMS ON METHODS FOR SUMMING DIVERGENT SERIES 5Before the 19th century divergent series were widely used by Euler and others, but often led to confusing and contra-dictory results. A major problem was Eulers idea that any divergent series should have a natural sum, without rstdening what is meant by the sum of a divergent series. Cauchy eventually gave a rigorous denition of the sum ofa (convergent) series, and for some time after this divergent series were mostly excluded from mathematics. Theyreappeared in 1886 with Poincar's work on asymptotic series. In 1890 Cesaro realized that one could give a rigorousdenition of the sum of some divergent series, and dened Cesaro summation. (This was not the rst use of Cesarosummation which was used implicitly by Frobenius in 1880; Cesaros key contribution was not the discovery of thismethod but his idea that one should give an explicit denition of the sum of a divergent series.)In the years afterCesaros paper several other mathematicians gave other denitions of the sum of a divergent series, though these arenot always compatible: dierent denitions can give dierent answers for the sum of the same divergent series, sowhen talking about the sum of a divergent series it is necessary to specify which summation method one is using.3.2 Theorems on methods for summing divergent seriesA summability method M is regular if it agrees with the actual limit on all convergent series. Such a result is calledan abelian theorem for M, from the prototypical Abels theorem.More interesting and in general more subtle arepartial converse results, called tauberian theorems, froma prototype proved by Alfred Tauber. Here partial conversemeans that if M sums the series , and some side-condition holds, then was convergent in the rst place; withoutany side condition such a result would say that M only summed convergent series (making it useless as a summationmethod for divergent series).The operator giving the sum of a convergent series is linear, and it follows from the HahnBanach theorem thatit may be extended to a summation method summing any series with bounded partial sums. This fact is not veryuseful in practice since there are many such extensions, inconsistent with each other, and also since proving suchoperators exist requires invoking the axiom of choice or its equivalents, such as Zorns lemma. They are thereforenonconstructive.The subject of divergent series, as a domain of mathematical analysis, is primarily concerned with explicit and naturaltechniques such as Abel summation, Cesro summation and Borel summation, and their relationships. The advent ofWieners tauberian theorem marked an epoch in the subject, introducing unexpected connections to Banach algebramethods in Fourier analysis.Summation of divergent series is also related to extrapolation methods and sequence transformations as numericaltechniques. Examples for such techniques are Pad approximants, Levin-type sequence transformations, and order-dependent mappings related to renormalization techniques for large-order perturbation theory in quantummechanics.3.3 Properties of summation methodsSummation methods usually concentrate on the sequence of partial sums of the series. While this sequence doesnot converge, we may often nd that when we take an average of larger and larger initial terms of the sequence, theaverage converges, and we can use this average instead of a limit to evaluate the sum of the series. So in evaluatinga = a0 + a1 + a2 + ..., we work with the sequence s, where s0 = a0 and sn = sn + an+1.In the convergent case,the sequence s approaches the limit a. A summation method can be seen as a function from a set of sequences ofpartial sums to values. If A is any summation method assigning values to a set of sequences, we may mechanicallytranslate this to a series-summation method A that assigns the same values to the corresponding series. There arecertain properties it is desirable for these methods to possess if they are to arrive at values corresponding to limitsand sums, respectively.1. Regularity. Asummation method is regular if, whenever the sequence s converges to x, A(s) = x. Equivalently,the corresponding series-summation method evaluates A(a) = x.2. Linearity. A is linear if it is a linear functional on the sequences where it is dened, so that A(k r + s) = kA(r) + A(s) for sequences r, s and a real or complex scalar k.Since the terms an = sn sn of the series aare linear functionals on the sequence s and vice versa, this is equivalent to A being a linear functional on theterms of the series.3. Stability (also called translativity). If s is a sequence starting froms0 and s is the sequence obtained by omittingthe rst value and subtracting it from the rest, so that sn = sn s0, then A(s) is dened if and only if A(s)6 CHAPTER 3. DIVERGENT SERIESis dened, and A(s) = s0 + A(s). Equivalently, whenever an = an for all n, then A(a) = a0 + A(a).[1][2]Another way of stating this is that the shift rule must be valid for the series that are summable by this method.The third condition is less important, and some signicant methods, such as Borel summation, do not possess it.[3]One can also give a weaker alternative to the last condition.1. Finite re-indexability. If a and a are two series such that there exists a bijection f: N N such that ai =af(i) for all i, and if there exists some N N such that ai = ai for all i > N, then A(a) = A(a). (In otherwords, a is the same series as a, with only nitely many terms re-indexed.) Note that this is a weaker conditionthan Stability, because any summation method that exhibits Stability also exhibits Finite Re-indexability,but the converse is not true.A desirable property for two distinct summation methods A and B to share is consistency:A and B are consistent iffor every sequence s to which both assign a value, A(s) = B(s).If two methods are consistent, and one sums moreseries than the other, the one summing more series is stronger.There are powerful numerical summation methods that are neither regular nor linear, for instance nonlinear sequencetransformations like Levin-type sequence transformations and Pad approximants, as well as the order-dependentmappings of perturbative series based on renormalization techniques.Taking regularity, linearity and stability as axioms, it is possible to sum many divergent series by elementary algebraicmanipulations. This partly explains why many dierent summation methods give the same answer for certain series.For instance, whenever r 1, the geometric seriesG(r, c) =k=0crk= c +k=0crk+1(stability)= c + rk=0crk(linearity)= c + r G(r, c), henceG(r, c) =c1 r, unless it is innitecan be evaluated regardless of convergence. More rigorously, any summation method that possesses these propertiesand which assigns a nite value to the geometric series must assign this value. However, when r is a real numberlarger than 1, the partial sums increase without bound, and averaging methods assign a limit of .3.4 Classical summation methodsThe two classical summation methods for series, ordinary convergence and absolute convergence, dene the sum asa limit of certain partial sums. Strictly speaking these are not really summation methods for divergent series, as bydenition a series is divergent only if these methods do not work, but are included for completeness. Most but not allsummation methods for divergent series extend these methods to a larger class of sequences.3.4.1 Absolute convergenceAbsolute convergence denes the sum of a sequence (or set) of numbers to be the limit of the net of all partial sumsak1+ ...+akn, if it exists. It does not depend on the order of the elements of the sequence, and a classical theorem saysthat a sequence is absolutely convergent if and only if the sequence of absolute values is convergent in the standardsense.3.5. NRLUND MEANS 73.4.2 Sum of a seriesCauchys classical denition of the sum of a series a0+a1+... denes the sum to be the limit of the sequence of partialsums a0+ ...+an. This is the default denition of convergence of a sequence.3.5 Nrlund meansSuppose pn is a sequence of positive terms, starting from p0. Suppose also thatpnp0 + p1 + + pn0.If now we transform a sequence s by using p to give weighted means, settingtm=pms0 + pm1s1 + + p0smp0 + p1 + + pmthen the limit of tn as n goes to innity is an average called the Nrlund mean Np(s).The Nrlund mean is regular, linear, and stable. Moreover, any two Nrlund means are consistent.3.5.1 Cesro summationThe most signicant of the Nrlund means are the Cesro sums. Here, if we dene the sequence pkbypkn=(n + k 1k 1)then the Cesro sum Ck is dened by Ck(s) = Npk(s). Cesro sums are Nrlund means if k 0, and hence areregular, linear, stable, and consistent. C0 is ordinary summation, and C1 is ordinary Cesro summation. Cesro sumshave the property that if h > k, then Ch is stronger than Ck.3.6 Abelian meansSuppose = {0, 1, 2, ...} is a strictly increasing sequence tending towards innity, and that 0 0. Supposef(x) =n=0an exp(nx)converges for all real numbers x>0. Then the Abelian mean A is dened asA(s) =limx0+f(x).More generally, if the series for f only converges for large x but can be analytically continued to all positive real x,then one can still dene the sum of the divergent series by the limit above.A series of this type is known as a generalized Dirichlet series; in applications to physics, this is known as the methodof heat-kernel regularization.Abelian means are regular and linear, but not stable and not always consistent between dierent choices of . However,some special cases are very important summation methods.8 CHAPTER 3. DIVERGENT SERIES3.6.1 Abel summationSee also: Abels theoremIf n = n, then we obtain the method of Abel summation. Heref(x) =n=0anenx=n=0anzn,where z = exp(x). Then the limit of (x) as x approaches 0 through positive reals is the limit of the power series for(z) as z approaches 1 from below through positive reals, and the Abel sum A(s) is dened asA(s) =limz1n=0anzn.Abel summation is interesting in part because it is consistent with but more powerful than Cesro summation:A(s)= Ck(s) whenever the latter is dened. The Abel sum is therefore regular, linear, stable, and consistent with Cesrosummation.3.6.2 Lindelf summationIf n = n log(n), then (indexing from one) we havef(x) = a1 + a222x+ a333x+ .Then L(s), the Lindelf sum (Volkov 2001), is the limit of (x) as x goes to zero. The Lindelf sum is a powerfulmethod when applied to power series among other applications, summing power series in the Mittag-Leer star.If g(z) is analytic in a disk around zero, and hence has a Maclaurin series G(z) with a positive radius of convergence,then L(G(z)) = g(z) in the Mittag-Leer star. Moreover, convergence to g(z) is uniform on compact subsets of thestar.3.7 Analytic continuationSeveral summation methods involve taking the value of an analytic continuation of a function.3.7.1 Analytic continuation of power seriesIf anxnconverges for small complex x and can be analytically continued along some path from x=0 to the point x=1,then the sum of the series can be dened to be the value at x=1. This value may depend on the choice of path.3.7.2 Euler summationMain article: Euler summationEuler summation is essentially an explicit formof analytic continuation. If a power series converges for small complexz and can be analytically continued to the open disk with diameter from 1/(q+1) to 1 and is continuous at 1, then itsvalue at is called the Euler or (E,q) sum of the series a0+.... Euler used it before analytic continuation was dened ingeneral, and gave explicit formulas for the power series of the analytic continuation.The operation of Euler summation can be repeated several times, and this is essentially equivalent to taking an analyticcontinuation of a power series to the point z=1.3.8. INTEGRAL FUNCTION MEANS 93.7.3 Analytic continuation of Dirichlet seriesThis method denes the sum of a series to be the value of the analytic continuation of the Dirichlet seriesf(s) =a11s+a22s+a33s+ at s=0, if this exists and is unique. This method is sometimes confused with zeta function regularization.3.7.4 Zeta function regularizationIf the seriesf(s) =1as1+1as2+1as3+ (for positive values of the an) converges for large real s and can be analytically continued along the real line tos=1, then its value at s=1 is called the zeta regularized sum of the series a1+a2+... Zeta function regularization isnonlinear. In applications, the numbers ai are sometimes the eigenvalues of a self-adjoint operator A with compactresolvent, and f(s) is then the trace of As. For example, if A has eigenvalues 1, 2, 3, ... then f(s) is the Riemann zetafunction, (s), whose value at s=1 is 1/12, assigning a value to the divergent series is 1 + 2 + 3 + 4 + .Othervalues of s can also be used to assign values for the divergent sums (0)=1 + 1 + 1 + ... = 1/2, (2)=1 + 4 + 9 +... = 0 and in general (s) =n=1 ns= 1s+ 2s+ 3s+ . . . = Bs+1s+1, where B is a Bernoulli number.[4]3.8 Integral function meansIf J(x)=pnxnis an integral function, then the J sum of the series a0+... is dened to belimxn pn(a0 + + an)xnn pnxn,if this limit exists.There is a variation of this method where the series for J has a nite radius of convergence r and diverges at x=r. Inthis case one denes the sum as above, except taking the limit as x tends to r rather than innity.3.8.1 Borel summationIn the special case when J(x)=exthis gives one (weak) form of Borel summation.3.8.2 Valirons methodValirons method is a generalization of Borel summation to certain more general integral functions J. Valiron showedthat under certain conditions it is equivalent to dening the sum of a series aslimn+H(n)2hZeh2H(n)/2(a0 + + ah)where H is the second derivative of G and c(n)=eG(n).10 CHAPTER 3. DIVERGENT SERIES3.9 Moment methodsSuppose that d is a measure on the real line such that all the momentsn=xndare nite. If a0+a1+... is a series such thata(x) =a0x00+a1x11+ converges for all x in the support of , then the (d) sum of the series is dened to be the value of the integrala(x)dif it is dened. (Note that if the numbers n increase too rapidly then they do not uniquely determine the measure .)3.9.1 Borel summationFor example, if d = exdx for positive x and 0 for negative x then n = n!, and this gives one version of Borelsummation, where the value of a sum is given by0et antnn!dt.There is a generalization of this depending on a variable , called the (B',) sum, where the sum of a series a0+... isdened to be0etantn(n + 1)dtif this integral exists. A further generalization is to replace the sum under the integral by its analytic continuationfrom small t.3.10 Miscellaneous methods3.10.1 Hausdor transformationsHardy (1949, chapter 11).3.10.2 Hlder summationMain article: Hlder summation3.10.3 Huttons methodIn 1812 Hutton introduced a method of summing divergent series by starting with the sequence of partial sums, andrepeated applying the operation of replacing a sequence s0, s1, ... by the sequence of averages (s0+ s1)/2, (s1+ s2)/2,..., and then taking the limit (Hardy 1949, p. 21).3.10. MISCELLANEOUS METHODS 113.10.4 Ingham summabilityThe series a1+... is called Ingham summable to s iflimx1nxannx[xn] = sAlbert Ingham showed that if is any positive number then (C,) (Cesaro) summability implies Ingham summa-bility, and Ingham summability implies (C,) summability Hardy (1949, Appendix II).3.10.5 Lambert summabilityThe series a1+... is called Lambert summable to s iflimy0+n1annyeny1 eny= sIf a series is (C,k) (Cesaro) summable for any k then it is Lambert summable to the same value, and if a series isLambert summable then it is Abel summable to the same value Hardy (1949, Appendix II).3.10.6 Le Roy summationThe series a0+... is called Le Roy summable to s iflim1n(1 + n)(1 + n)an= sHardy (1949, 4.11)3.10.7 Mittag-Leer summationThe series a0+... is called Mittag-Leer (M) summable to s iflim0nan(1 + n)= sHardy (1949, 4.11)3.10.8 Ramanujan summationMain article: Ramanujan summationRamanujan summation is a method of assigning a value to divergent series used by Ramanujan and based on theEulerMaclaurin summation formula. The Ramanujan sum of a series f(0) + f(1) + ... depends not only on thevalues of f at integers, but also on values of the function f at non-integral points, so it is not really a summationmethod in the sense of this article.3.10.9 Riemann summabilityThe series a1+... is called (R,k) (or Riemann) summable to s if12 CHAPTER 3. DIVERGENT SERIESlimh0nan(sin nhnh)k= sHardy (1949, 4.17). The series a1+... is called R2 summable to s iflimh02nsin2nhn2h(a1 + an) = s3.10.10 Riesz meansMain article: Riesz meanIf n form an increasing sequence of real numbers andA(x) = a0 + + an for n< x n+1then the Riesz (R,,) sum of the series a0+... is dened to belim0A(x)( x)1dx3.10.11 Valle-Poussin summabilityThe series a1+... is called VP (or Valle-Poussin) summable to s iflimma0 + a1mm + 1+ a2m(m1)(m + 1)(m + 2)+ = sHardy (1949, 4.17).3.11 See also1 1 + 2 6 + 24 120 + SilvermanToeplitz theorem3.12 Notes[1] see Michons Numericana http://www.numericana.com/answer/sums.htm[2] see also Translativity at The Encyclopedia of Mathematics wiki (Springer)[3] Muraev, . B. (1978), Borel summation of n-multiple series, and entire functions associated with them,AkademiyaNauk SSSR 19 (6): 13321340, 1438, MR 515185. Muraev observes that Borel summation is translative in one of thetwo directions:augmenting a series by a zero placed at its start does not change the summability or value of the series.However, he states the converse is false.[4] Tao, Terence (10 April 2010). The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variableanalytic continuation.3.13. REFERENCES 133.13 ReferencesArteca, G.A.; Fernndez, F.M.; Castro, E.A. (1990), Large-Order Perturbation Theory and Summation Meth-ods in Quantum Mechanics, Berlin: Springer-Verlag.Baker, Jr., G. A.; Graves-Morris, P. (1996), Pad Approximants, Cambridge University Press.Brezinski, C.; Zaglia, M. Redivo (1991), Extrapolation Methods. Theory and Practice, North-Holland.Hardy, G. H. (1949), Divergent Series, Oxford: Clarendon Press.LeGuillou, J.-C.; Zinn-Justin, J. (1990), Large-Order Behaviour of Perturbation Theory, Amsterdam: North-Holland.Volkov, I.I. (2001), Lindelf summation method, in Hazewinkel, Michiel,Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4.Zakharov, A.A. (2001), Abel summation method, in Hazewinkel, Michiel, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4.Hazewinkel, Michiel, ed. (2001), Riesz summation method, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-414 CHAPTER 3. DIVERGENT SERIES3.14 Text and image sources, contributors, and licenses3.14.1 Text Darbouxs formula Source: https://en.wikipedia.org/wiki/Darboux{}s_formula?oldid=607144469 Contributors: Michael Hardy, Bearcat,Giftlite, Xezbeth, R.e.b., RussBot, JJL, SmackBot, Berland, A. Pichler, David Eppstein, Bearian, Yobot and Anonymous: 2 Dimensional regularizationSource: https://en.wikipedia.org/wiki/Dimensional_regularization?oldid=639331453 Contributors: MichaelHardy, Phys, Lumidek, David Schaich, Xezbeth, Flammifer, Fwb22, Ronark, Rjwilmsi, R.e.b., Chobot, Conscious, Zunaid, SmackBot,Siebren, Silly rabbit, TriTertButoxy, Headbomb, Shambolic Entity, Natsirtguy, Mild Bill Hiccup, Addbot, Niout, AnomieBOT, Xqbot,Omnipaedista, Erik9bot, Fcametti, Naviguessor, Suslindisambiguator, Bibcode Bot, AEIOU29979 and Anonymous: 17 DivergentseriesSource: https://en.wikipedia.org/wiki/Divergent_series?oldid=670866416Contributors: AxelBoldt, XJaM, MichaelHardy, Chinju, Charles Matthews, Jitse Niesen, Hyacinth, Gandalf61, MathMartin, Tobias Bergemann, Giftlite, Gene Ward Smith,Waltpohl, Meddlin' Pedant, Rgdboer, Crislax, Dfeldmann, Msh210, Mrholybrain, Linas, Sympleko, Tygar, Triddle, NatusRoma, Dan-wbartlett, Salix alba, R.e.b., LMSchmitt, Dppowell, DerHannes, Kier07, Attilios, Melchoir, Dan Hoey, MalafayaBot, Silly rabbit, Daqu,Mgiganteus1, Jim.belk, Dchudz, ChrisCork, Bons, Mudd1, Gregbard, Ntsimp, Arturocl, Mglg, Oerjan, Dawnseeker2000, Albmont,David Eppstein, Tercer, F3et, Policron, VolkovBot, LokiClock, AlleborgoBot, Qwfp, DumZiBoT, MystBot, Addbot, Nordisk varg, Zor-robot, Ettrig, Luckas-bot, Xqbot, Drilnoth, FrescoBot, Sawomir Biay, Kiefer.Wolfowitz, Asllearner, Ali Abbasi7, FoxBot, Numericana,ClueBot NG, Guy vandegrift, Trevayne08, Brad7777, BattyBot, Pawe Ziemian, Kanghuitari, Faizan, DilatoryRevolution, TCMemoire,A.bt(w) and Anonymous: 453.14.2 Images File:Fibonacci_spiral_34.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/93/Fibonacci_spiral_34.svg License: Publicdomain Contributors: self-drawn in Inkscape Original artist: User:Dicklyon3.14.3 Content license Creative Commons Attribution-Share Alike 3.0