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Convergence Acceleration of Series Through a Variational Approach

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Convergence Acceleration of Series Through a Variational Approach Convergence acceleration of series through a variational approach September 30, 2004 Paolo Amore∗, Facultad de Ciencias, Universidad de Colima, Bernal D´ıaz del Castillo 340, Colima, Colima, M´exico Abstract By means of a variational approach we find new series representa- tions both for well known mathematical constants, such as π and the Catalan constant, and for mathematical functions, such as the Rie- mann zeta function. The series that we have found are all exponen- tially convergent and provide quite useful analytical approximations. With limited effort our method can be applied to obtain similar expo- nentially convergent series for a large class of mathematical functions. 1 Introduction This paper deals with the problem of improving the convergence of a series in the case where the series itself is converging very slowly and a large number of terms needs to be evaluated to reach the desired accuracy. This is an interesting and challenging problem, which has been considered by many authors before (see, for example, [FV 1996, Br 1999, Co 2000]). In this paper we wish to attack this problem, although aiming from a different angle: instead of looking for series expansions in some small nat- ural parameter (i.e. a parameter present in the original formula), we in- troduce an artificial dependence in the formulas upon an (not completely) ∗[email protected] 1 arbitrary parameter and then devise an expansion which can be optimized to give faster rates of convergence. The details of how this work will be explained in depth in the next section. This procedure is well known in Physics and it has been exploited in the so-called \Linear Delta Expansion" (LDE) and similar approaches [AFC 1990, Jo 1995, Fe 2000]. The author himself has devised different methods which allow to treat a large class of physical/mathematical problems: the non-perturbative generalization of the Lindstedt-Poincar´e method [AA 2003a, AA 2003b, AM 2004], the solution of the Schr¨odinger equation in quantum mechanics [Am 2004b, Am 2004c], the calculation of a certain class of integrals of physical interest [AS 2004, Am 2004a] and the analytical approximation of WKB expansion in quantum mechanics [AL 2004]. In the following we want to show that such variational techniques can also be used to obtain exponentially convergent series for some mathematical functions. Although we will focus on a few functions, the method that we apply is quite flexible and can be certainly applied to a larger number of examples. 2 Mathematical constants Many fundamental mathematical constants can be expressed as infinite sums. In many cases such series converge very slowly and a huge number of terms has to be counted before reaching the desired precision. Several examples of this are discussed for example in [FV 1996], where the authors consider a par- ticular rearrangement of the series which transforms them into rapidly con- verging series. In the following we review two of the examples of [FV 1996], π and the Catalan constant, and apply a variational method to further improve the convergence rate of these sums. 2.1 The number π The series 1 1 1 S = 4 − (1) 4n − 3 4n − 1 n=1 X converges very slowly to π and it is known as Gregory's formula. The sum of the to the first 103 terms yields an approximate value of π, which only has 2 the first 3 decimals correct. As discussed in [FV 1996] it is possible to convert series such as the one in eq. (1) into rapidly converging ones. In the following we generalize the method of Flajolet and Vardi to make room for a variational parameter. The basic idea is to introduce in the series an arbitrary parameter, which is then tuned to accelerate the convergence of the series itself. The principle of minimal sensitivity (PMS) [Ste 1981] is used to fix the value of the parameter. Let us see how this works: we write the series as 1 1 1 1 1 S = 3 − 1 : (2) n 1 + λ 0 4n +λ 4n +λ 1 n=1 1 − 1 − X 1+λ 1+λ 3 @1 A 4n +λ 4n +λ Provided that 1+λ < 1 and 1+λ < 1, i.e. λ > −5=8, we can expand eq. (2) as: 1 1 m m 1 1 3 + λ 1 + λ S = 4 4n − 4n 4n 1 + λ 1 + λ 1 + λ n=1 m=1 X1 1 X m m k k 1 1 +1 m 3 1 = 4 λm−k − 4n 1 + λ k 4n 4n n=1 m=1 k=1 " # X1 X m m X 1 +1 m 3k − 1 = 4 λm−k ζ(k + 1) ; (3) 1 + λ k 4k+1 m=1 k X X=1 where the last line has been obtained by performing the summation over n. For λ = 0 eq. (3) yields the expression found in [FV 1996], i.e. 1 3m − 1 S(F V ) = ζ(m + 1) ; (4) 4m m=1 X which converges exponentially to π. Let us go back to eq. (3). This series converges to π for any value of λ > −5=8: as a result, we can regard the dependence upon λ of the partial sums of the first N terms of eq. (3) as artificial, since it is bound to disappear in the limit N ! 1. We therefore minimize such dependence using the PMS and requiring that the derivative of the equation with respect to λ vanish. To leading order, which corresponds to keeping only two terms in the sum, we obtain the optimal value of λ: 3 λ(1) = − ζ(3) ≈ −0:365381 > −5=8 : (5) π2 3 In Fig. 1 we display the partial sums of eq. (3) with 10, 20 and 30 terms as a function of λ: the locations of the optimal value (to first order) and λ = 0 are marked with vertical lines. It turns out that λ(1) is an excellent approximation to the exact minimum of the partial sum even for large values of terms. In Fig. 2 we plot the error obtained by using eq. (3) with λ given by eq. (5) and by using the formula of Flajolet and Vardi, eq. (4). Our series converges exponentially more rapidly than eq. (4). 2.2 The Catalan constant The series 1 1 1 S = − (6) (4n − 3)2 (4n − 1)2 n=1 X is known to slowly converge to the Catalan constant, G ≈ 0:9159656. We rewrite it as d S = lim S~(a) (7) a!0 da where 1 1 1 S~(a) ≡ − (4n − 3 − a) (4n − 1 − a) n=1 X1 1 m m 1 1 3+a + λ 1+a + λ = 4n − 4n 4n 1 + λ 1 + λ 1 + λ n=1 m=1 X1 1 X m m k k 1 1 +1 m 3 + a 1 + a = λm−k − 4n 1 + λ k 4n 4n n=1 m=1 k=1 " # X1 X m m X 1 +1 m (3 + a)k − (1 + a)k = λm−k ζ(k + 1) : (8) 1 + λ k 4k+1 m=1 k X X=1 In principle we would be tempted to perform the derivation with respect to a and then take the limit a ! 0, thus obtaining the expression 1 m m 1 +1 m 3k−1 − 1 S = λm−k k ζ(k + 1) ; (9) 1 + λ k 4k+1 m=1 k X X=1 4 which indeed for λ = 0 reduces to the formula given in [FV 1996]: 1 3m−1 − 1 S = m ζ(m + 1) : (10) 4m+1 m=1 X However in doing so we have neglected that λ can be function of a and therefore we expect that a better approach will consist in first applying the PMS to S~ and then take the limit. In such a case the PMS to leading order yields the optimal value 3 a λ(1) = − ζ(3) 1 + : (11) π2 2 By using this value we obtain 1 m m−k 1 1 λ0 m k S = 3+2k m+2 ζ(k + 1) − 3 + 3 k (1 + λ0) 3 2 (1 + λ0) k m=1 k=1 X X k − 3 −1 + 3 (λ0 − m) (12) 3 where we have defined λ0 ≡ −π2 ζ(3). In Fig. 3 we compare the different approximations, showing that eq. (9) (with λ = λ(1)) and eq. (12) have a greater rate of convergence then the corresponding equation of [FV 1996]. Eq. (12) provides a slightly better approximation. The results derived above can be generalized to sums of the form: 1 1 1 S = − (13) n (4n − 3)n (4n − 1)n n=1 X simply using 1 dn−1 Sn = lim S~(a): (14) (n − 1)! a!0 dan−1 3 The Riemann zeta function We can apply the same strategy outlined above to the calculation of the Riemann zeta function. We consider the integral representation (−2)n−1 1 1 logn−1 x ζ(n) = dx (15) 2n−1 − 1 Γ(n) 1 + x Z0 5 and write it as: (−2)n−1 1 1 1 logn−1 x ζ(n) = dx; (16) 2n−1 − 1 Γ(n) 1 + λ 1 + x−λ Z0 1+λ where λ is as usual an arbitrary parameter introduced by hand. x−λ The condition 1+λ < 1 is fullfilled provided that λ > 0; in this case one can expand the denominator in powers of x−λ and obtain: 1+λ 1 (−2)n−1 1 1 1 ζ(n) = (−x + λ)k logn−1 x dx 2n−1 − 1 Γ(n) (1 + λ)k+1 k=0 Z0 X1 k (−2)n−1 1 1 k! 1 = λk−j(−1)j xj logn−1 x 2n−1 − 1 Γ(n) (1 + λ)k+1 j!(k − j)! k j=0 0 X=0 X Z 1 k (−2)n−1 1 1 k! Γ(n) = λk−j(−1)n+j+1 2n−1 − 1 Γ(n) (1 + λ)k+1 j!(k − j)! (1 + j)n k j=0 X=0 X 1 k (2)n−1 1 k! (−1)j = λk−j : (17) 2n−1 − 1 (1 + λ)k+1 j!(k − j)! (1 + j)n k j=0 X=0 X Although λ appears explicitly in the series (17), the series itself does not depend upon λ, as long as λ > 0.
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