Summability methods eFrom Wikipedia, the free encyclopediaContents1 Euler summation 11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 EulerMaclaurin formula 32.1 The formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 The Bernoulli polynomials and periodic function. . . . . . . . . . . . . . . . . . . . . . . 42.1.2 The remainder term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Applicable formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 The Basel problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Sums involving a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.3 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.4 Asymptotic expansion of sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.1 Derivation by mathematical induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9iChapter 1Euler summationIn the mathematics of convergent and divergent series, Euler summation is a summability method. That is, it is amethod for assigning a value to a series, dierent fromthe conventional method of taking limits of partial sums. Givena series an, 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 dene values for divergent series, Euler summation can be used to speed the convergence ofseries.Euler summation can be generalized into a family of methods denoted (E, q), where q 0. The (E, 0) sum is theusual (convergent) sum, while (E, 1) is the ordinary Euler sum. All of these methods are strictly weaker than Borelsummation; for q > 0 they are incomparable with Abel summation.1.1 DenitionEuler summation is particularly used to accelerate the convergence of alternating series and allows evaluating divergentsums.Eyj=0aj:=i=01(1 + y)i+1ij=0_ij_yj+1aj.To justify the approach notice that for interchanged sum, Eulers summation reduces to the initial series, becauseyj+1i=j_ij_1(1 + y)i+1= 1.This method itself cannot be improved by iterated application, asEy1Ey2 =E y1y21+y1+y2.1.2 ExamplesWe have j=0 xjPk(j) =ki=0xi(1x)i+1ij=0_ij_(1)ijPk(j) , if Pk is a polynomial of degree k. Notethat in this case Euler summation reduces an innite series to a nite sum.The particular choice Pk(j):=(j + 1)kprovides an explicit representation of the Bernoulli numbers, since(k) = Bk+1k+1. Indeed, applying Euler summation to the zeta function yields112k+1ki=012i+1ij=0_ij_(1)j(j+1)k, which is polynomial for k a positive integer; cf. Riemann zeta function.12 CHAPTER 1. EULER SUMMATION j=0 zj=i=01(1+y)i+1ij=0_ij_yj+1zj=y1+yi=0_1+yz1+y_i. With an appropriate choice of y thisseries converges to11z .1.3 See alsoBorel summationCesro summationLambert summationPerrons formulaAbelian and tauberian theoremsAbelPlana formulaAbels summation formulaVan Wijngaarden transformation1.4 ReferencesKorevaar, Jacob (2004). Tauberian Theory: A Century of Developments. Springer. ISBN 3-540-21058-X.Shawyer, Bruce and Bruce Watson (1994). Borels Methods of Summability: Theory and Applications. OxfordUP. ISBN 0-19-853585-6.Apostol, Tom M. (1974). Mathematical Analysis Second Edition. Addison Wesley Longman. ISBN 0-201-00288-4.Chapter 2EulerMaclaurin formulaIn mathematics, theEulerMaclaurinformula provides a powerful connection between integrals (see calculus)and sums. It can be used to approximate integrals by nite sums, or conversely to evaluate nite sums and inniteseries using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from theformula, and Faulhabers formula for the sum of powers is an immediate consequence.The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735 (and later general-ized as Darbouxs formula). Euler needed it to compute slowly converging innite series while Maclaurin used it tocalculate integrals.2.1 The formulaIf m and n are natural numbers and f ( x ) is a complex or real valued continuous function for real numbers x in theinterval [m, n] , then the integralI=nmf(x) dxcan be approximated by the sum (or vice versa)S= f (m + 1) + + f (n 1) + f(n)(see trapezoidal rule). The EulerMaclaurin formula provides expressions for the dierence between the sum and theintegral in terms of the higher derivatives f ( k ) ( x ) evaluated at the end points of the interval, that is to say when x=m and n.Explicitly, for a natural number p and a function f(x) that is p times continuously dierentiable in the interval [m, n], we haveS I=pk=1Bkk!_f(k1)(n) f(k1)(m)_ + Rwhere Bk is the kth Bernoulli number, with B0 = 1, B1 = +1/2, B2 = 1/6, B3 = 0, B4 = 1/30, B5 = 0, B6 = 1/42, B7= 0, B8 = 1/30 ..., and R is an error term which is normally small for suitable values of p and depends on n, m, p andf.The formula is often written with the subscript taking only even values, since the odd Bernoulli numbers are zeroexcept for B1, in which case we haveni=m+1f(i) =nmf(x) dx + B1 (f(n) f(m)) +pk=1B2k(2k)!_f(2k1)(n) f(2k1)(m)_ + R.34 CHAPTER 2. EULERMACLAURIN FORMULA2.1.1 The Bernoulli polynomials and periodic functionThe formula is derived below using repeated integration by parts applied to successive intervals [r, r +1] for integersr= m, m + 1, , n 1 . The derivation uses the periodic Bernoulli functions, Pk(x) which are dened in termsof the Bernoulli polynomials Bk(x) for k = 0, 1, 2 .the Bernoulli polynomials may be dened recursively byB0(x) = 1Bn(x) = nBn1(x) and10Bn(x) dx = 0 for n 1and the periodic Bernoulli functions are dened asPn(x) = Bn (x x)where x denotes the largest integer that is not greater than x so that x - x always lies in the interval [ 0, 1 ].It can be shown that Bk(1) = Bk(0) for all k = 1 so that except for P1(x) , all the periodic Bernoulli functions arecontinuous. The functions Pk(x) are sometimes written asBk(x) .2.1.2 The remainder termThe remainder term R can be written asR =nmf(2p)(x)P2p(x)(2p)!dxor equivalently, integrating by parts, assuming (2p) is dierentiable again and recalling that all odd Bernoulli numbers(but the rst one) are zero:R = nmf(2p+1)(x)P(2p+1)(x)(2p + 1)!dx p > 0When n > 0, it can be shown that|Bn (x)| 2 n!(2)n (n)where denotes the Riemann zeta function; one approach to prove this inequality is to obtain the Fourier series forthe polynomials Bn ( x ). The bound is achieved for even n when x is zero. The term (n) may be omitted for odd nbut the proof in this case is more complex (see Lehmer:[1])Using this inequality, the size of the remainder term can be estimated using|R| 2(2p)(2)2pnmf(2p)(x) dx2.1.3 Applicable formulaWe can use the formula as a means of approximating a nite integral, with the following simple formula:[2]I=xNx0f(x) dx = h_f02+ f1 + f2... + fN1 +fN2_ +h212[f0 fN] h4720[f0 fN ] + ...2.2. APPLICATIONS 5Where N is the number of points in the interval of integration from x0 to xN and h is the distance between pointsso that h=(xN x0)/N .Note the series above is usually not convergent; indeed, often the terms will increaserapidly after a number of iterations. Thus, attention generally needs to be paid to the remainder term.This may be viewed as an extension of the trapezoid rule by the inclusion of correction terms.[2]2.2 Applications2.2.1 The Basel problemThe Basel problem asks to determine the sum1 +14+19+116+125+ =n=11n2Euler computed this sum to 20 decimal places with only a few terms of the EulerMaclaurin formula in 1735. Thisprobably convinced him that the sum equals 2/ 6, which he proved in the same year.[3] Parsevals identity for theFourier series of f(x) = x gives the same result.2.2.2 Sums involving a polynomialIf f is a polynomial and p is big enough, then the remainder term vanishes. For instance, if f(x) = x3, we can choosep = 2 to obtain after simplicationni=0i3=_n(n + 1)2_2(see Faulhabers formula).2.2.3 Numerical integrationThe EulerMaclaurin formula is also used for detailed error analysis in numerical quadrature. It explains the supe-rior performance of the trapezoidal rule on smooth periodic functions and is used in certain extrapolation methods.ClenshawCurtis quadrature is essentially a change of variables to cast an arbitrary integral in terms of integrals ofperiodic functions where the EulerMaclaurin approach is very accurate (in that particular case the EulerMaclaurinformula takes the form of a discrete cosine transform). This technique is known as a periodizing transformation.2.2.4 Asymptotic expansion of sumsIn the context of computing asymptotic expansions of sums and series, usually the most useful form of the EulerMaclaurin formula isbn=af(n) baf(x) dx +f(b) + f(a)2+k=1B2k(2k)!_f(2k1)(b) f(2k1)(a)_where a and b are integers.[4] Often the expansion remains valid even after taking the limits aor b+, or both.In many cases the integral on the right-hand side can be evaluated in closed form in terms of elementary functionseven though the sum on the left-hand side cannot. Then all the terms in the asymptotic series can be expressed interms of elementary functions. For example,6 CHAPTER 2. EULERMACLAURIN FORMULAk=01(z + k)2 01(z + k)2dk. .=1z+12z2+t=1B2tz2t+1Here the left-hand side is equal to (1)(z) , namely the rst-order polygamma function dened through (1)(z)=d2dz2 ln (z); the gamma function (z) is equal to (z1)! if z is a positive integer. This results in an asymptotic expansion for (1)(z). That expansion, in turn, serves as the starting point for one of the derivations of precise error estimates for Stirlingsapproximation of the factorial function.2.2.5 Examples nk=11ks=1ns1+ s n1xxs+1dx with s R \ {1} nk=11k= log n +12+12n n1xxx2dx2.3 Proofs2.3.1 Derivation by mathematical inductionWe follow the argument given in Apostol.[5]The Bernoulli polynomials Bn(x) and the periodic Bernoulli functions Pn(x) for n = 0, 1 ,2, ... were introduced above.The rst several Bernoulli polynomials areB1(x) = x 12B2(x) = x2 x +16B3(x) = x332x2+12xB4(x) = x4 2x3+ x2130...The values Bn(0) are the Bernoulli numbers. Notice that for n 1 we haveBn(0) = Bn(1) = Bn(nnumber Bernoulli th)For n = 1,B1(0) = B1(1) = B1The functions Pn agree with the Bernoulli polynomials on the interval [ 0, 1 ] and are periodic with period 1. Fur-thermore, except when n = 1, they are also continuous. Thus,Pn(0) = Pn(1) = Bn, n = 1Let k be an integer, and consider the integral2.3. PROOFS 7k+1kf(x) dx =k+1kudvwhereu = f(x)du = f(x) dxdv= P0(x) dx sinceP0(x) = 1v= P1(x)Integrating by parts, we getk+1kf(x) dx =_uv_k+1kk+1kv du=_f(x)P1(x)_k+1kk+1kf(x)P1(x) dx= B1(1)f(k + 1) B1(0)f(k) k+1kf(x)P1(x) dxSumming the above from k = 0 to k = n 1, we get10f(x) dx + +nn1f(x) dx =n0f(x) dx=f(0)2+ f(1) + + f(n 1) +f(n)2n0f(x)P1(x) dxAdding (f(n) - f(0))/2 to both sides and rearranging, we havenk=1f(k) =n0f(x) dx +f(n) f(0)2+n0f(x)P1(x) dx (1)The last two terms therefore give the error when the integral is taken to approximate the sum.Next, considerk+1kf(x)P1(x) dx =k+1kudvwhereu = f(x)du = f(x) dxdv= P1(x) dxv=12P2(x)Integrating by parts again, we get_uv_k+1kk+1kv du =_f(x)P2(x)2_k+1k12k+1kf(x)P2(x) dx=B22(f(k + 1) f(k)) 12k+1kf(x)P2(x) dx8 CHAPTER 2. EULERMACLAURIN FORMULAThen summing from k = 0 to k = n 1, and then replacing the last integral in (1) with what we have thus shown tobe equal to it, we havenk=1f(k) =n0f(x) dx +f(n) f(0)2+B22(f(n) f(0)) 12n0f(x)P2(x) dx.By now the reader will have guessed that this process can be iterated. In this way we get a proof of the EulerMaclaurin summation formula which can be formalized by mathematical induction, in which the induction step relieson integration by parts and on the identities for periodic Bernoulli functions.2.4 See alsoCesro summationEuler summationGaussKronrod quadrature formulaDarbouxs formula2.5 Notes[1] Lehmer, D. H. (1940). On the maxima and minima of Bernoulli polynomials. The American Mathematical Monthly 47(8): 533538.[2] Devries, Paul L.; Hasbrun, Javier E. (2011). Arst course in computational physics. (2nd ed.). Jones and Bartlett Publishers.p. 156.[3] Pengelley, David J. Dances between continuous and discrete:Eulers summation formula, in:Robert Bradley and EdSandifer (Eds), Proceedings, Euler 2K+2 Conference (Rumford, Maine, 2002), Euler Society, 2003.[4] Abramowitz & Stegun (1972), 23.1.30[5] Apostol, T. M. (1 May 1999). An Elementary Viewof Eulers Summation Formula. The American Mathematical Monthly(Mathematical Association of America) 106 (5): 409418. doi:10.2307/2589145. ISSN 0002-9890. JSTOR 2589145.2.6 ReferencesAbramowitz, Milton; Stegun, Irene A., eds. (1972). Handbook of Mathematical Functions with Formulas,Graphs, and Mathematical Tables. New York: Dover Publications. ISBN 978-0-486-61272-0. tenth printing.,pp. 16, 806, 886Weisstein, Eric W., EulerMaclaurin Integration Formulas, MathWorld.Gourdon, Xavier; Sebah, Pascal Introduction on Bernoullis numbers, (2002)Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative number theory I. Classical theory. Cam-bridge tracts in advanced mathematics 97. pp. 495519. ISBN 0-521-84903-9.2.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 92.7 Text and image sources, contributors, and licenses2.7.1 Text Euler summation Source: https://en.wikipedia.org/wiki/Euler_summation?oldid=668367596 Contributors: Michael Hardy, Stevenj, An-drewman327, Giftlite, Crislax, Linas, Gadget850, Melchoir, Chris the speller, A. Pichler, Bons, CBM, Myasuda, Gromgull, DavidEppstein, TXiKiBoT, Yugsdrawkcabeht, Addbot, Ginosbot, Luckas-bot, Xqbot, Ripchip Bot, Slawekb, RealzGirlz and Anonymous: 7 EulerMaclaurin formula Source: https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula?oldid=663051876 Contribu-tors: AxelBoldt, Miguel~enwiki, RTC, Michael Hardy, Stevenj, Charles Matthews, Jitse Niesen, McKay, Robbot, Fredrik, Sverdrup,Giftlite, MuDavid, Crislax, Eric Kvaalen, Linas, Graham87, BD2412, Godzatswing, FlaBot, Mathbot, Bgwhite, Zarel, Merosonox,Brian Tvedt, JJL, SmackBot, RDBury, Kurykh, AdamSmithee, Berland, Giganut, Antares784, Beetstra, A. Pichler, Thijs!bot, Unifey~enwiki,.anacondabot, Magioladitis, Vanish2, Email4mobile, David Eppstein, VectorBundle, LokiClock, Sapphic, Logan, SieBot, Reuqr, Brewsohare, Kiensvay, Kbdankbot, Cuaxdon, Haklo, Lightbot, Legobot, Luckas-bot, Yobot, Angry bee, Citation bot, Xqbot, DSisyphBot, Fres-coBot, Citation bot 1, Shawnarchy, Tal physdancer, Achim1999, Graroo, Ripchip Bot, WilliamADon, ClueBot NG, Hjilderda, BG19bot,IluvatarBot, Michael.a.cohen, Brirush, Tentinator, Comp.arch, Nigellwh, Someone not using his real name, Monkbot, BruceMathSoft-wareGuy, Jorge Guerra Pires and Anonymous: 552.7.2 Images File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg License:Public domain Contributors:Created by bdesham with Inkscape; based upon Text-x-generic.svgfrom the Tango project. Original artist: Benjamin D. Esham (bdesham)2.7.3 Content license Creative Commons Attribution-Share Alike 3.0