Wallis sequence estimated through the Euler Maclaurin formula: even from the Wallis product π could be computed fairly accurately

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1 38 Wallis sequece estimated through the Euler Maclauri formula: eve from the Wallis product π could be computed fairly accurately Vito Lampret Summary The power of the Euler Maclauri summatio formula is illustrated through the example i which π is computed quite accurately from the slowly coverget Wallis sequece W := 4k 4k. Usig the Euler Maclauri formula the rate of covergece of 3 W is estimated as 0 < π W < 4, for iteger 3. Preseted example 0 implicitly suggests that perhaps i the udergraduate mathematics curriculum the Euler Maclauri formula of a lower order should be icluded, or simply replacig the Simpso s formula. Key words: acceleratio of covergece, Euler Maclauri summatio, the rate of covergece, Wallis product. MSC: 40A05, 40A0, 40A5, 65B0, 65B5. Itroductio The first presetatio of π := circumferece of a circle diameter of a circle i a form of a limit has bee made by Wallis i 655, see [ [4. This presetatio ca be obtaied i the followig way. For the sequece I k := π 0 si k x dx we have I 0 = π, I =,, usig the method of itegratio by parts, I k = k I k for k. k Cosequetly, puttig k = k = + N, we fid, by iductio, the ext two expressios Because I = I + = k k π k k +. 0 < si + x < si + x < si x < for x 0, π N, Joh Wallis, , Eglish mathematicia

2 Wallis sequece estimated through the Euler MacLauri formula 39 we have 0 < I + < I + < I < for all N. Hece, due to the previous expressios for I I +, the estimate holds. Therefore or + k k k k 4k 4k π < k k + < k k k k + < π + < k k < π < 4k 4k π k k for every N. Thus, the sequece 4k W := 4k k, + k ivolvig oly ratioal umbers, coverges towards the ifiite Wallis product W := lim W = 4k 4k = = π k + ; a see [, p. 58, [, p. 384,[7, p. 3, [4, pp. 5&47, [6, p. 384, [9, pp. 4&465 or [. This old iterestig formula seems to be of small practical value for umerical computatio of π as it is frequetly oted i the calculus textbooks. I fact, the Wallis sequece coverges very slowly its late terms are directly ot easily computable. However, eve from the Wallis product, π could be computed fairly accurately. Ideed, expressig the Wallis product by the Gamma fuctio [, pp. 55&58 as W = π Γ + + Γ + = π Γ + Γ + the usig the cotiuous versio of Stirlig s formula [, p. 57 π x x Θx Γx = x e x for x > 0 some Θ x 0,, we obtai the expressio W = πe + Θ exp + 6 Θ cosequetly also π = W + e + Θ exp Θ 6 for every positive iteger some Θ, Θ 0,. From this last formula it is possible to compute π to several decimal places from precedig formula we ca compute Wallis products also for large values of. The costat C figured i the origial versio of Stirlig s formula, Γx = C x x e x Θx x, was i fact determied just usig the Wallis product as C = π. This expressio for C was still ukow to Stirlig. b c

3 330 Vito Lampret I the lies above we have used two importat theorems cocerig Gamma fuctio, the last oe was the Stirlig s formula, which plays the crucial role i the approximatios give above. However, this theorem ca be deduced usig special techique of summatio kow as the Euler Maclauri summatio formula, see for example [,, 3, 4, 6, 7, 5. Although this formula is very importat, it is ot icluded, cotrary to our opiio, i the udergraduate curriculum. Perhaps due to the fact that may a mathematicia believes that the Euler Maclauri formula, as well as its derivatio, is too complicated to eter ito the udergraduate curriculum. Is it true? Certaily ot, as was clearly show i [7 [8. Moreover, usig oly the Euler Maclauri formula of order 3, the Wallis sequece, as well as π, could be estimated directly rather well, similarly to b c, avoidig all the complicatios quoted above. Hece, we wish to preset the power of the Euler Maclauri tool simultaeously complemet the article [7 with this example. Usig this formula we ca ofte successfully estimate defiite itegrals, sums products as well. I the preset article we use also the Taylor s formula, which is also deduced i [7. We wish to poit out to teachers what powerful device they are overlookig. We plead for the Euler Maclauri formula of order p 4 to be icluded ito the uiversities/colleges udergraduate curriculum. The absece of this formula i the mathematics curriculum is similar to the absece of DNA techology i foresic sciece. Namely, as it is possible to deduce from oly oe hair, foud o the victim, a lot of crucial coclusios, similarly we ca i some cases compute the sum of a slowly coverget series usig the Euler Maclauri formula, eve much more, see [7. I the preset article we illustrate this fact, where we trasform the slowly coverget Wallis sequece ito the faster coverget sequece with limit equal to π. This way we itroduce the umerical summatio techique to a wide commuity of readers. The preseted techique should attract teacher/studet ito a sphere of costructive cocrete mathematics. Derivatio of the Euler Maclauri formula of low order, for example of order p 4, ca be carried out quite easily, eve more easily tha it was doe i [7; see also [8. But with such a formula we ca compare itegral its itegral sum, which is iterestig for a teacher for a studet as well. For example, from 6, 7a, 7b 0 below we extract the Euler Maclauri formula of order p = 3 : k=m fk m fx dx = [f fm + [f f m 6 m P 3 x f 3 x dx, where P 3 is -periodic, differetiable fuctio, bouded as P 3 x < 0 for x R. So, if f 3 x is small, we ca make useful estimate of the differece betwee the itegral its itegral sum. This way we produce formulas for umerical itegratio summatio simultaeously. It is iterestig that the Euler Maclauri formula of order 4 gives for the absolute value of the remaider i the umerical itegratio rule Hermite s rule four times smaller a priori estimate as it is kow for the Simpso s rule. Moreover, Simpso s rule is less suitable for umerical summatio as compared to Hermite s rule [8. As a matter of fact the purpose of this article is ot the computatio of π, sice may very efficiet techiques for this task are kow. The mai purpose of this article is to itroduce the method, which trasforms practically useless formula ito applicable oe. Further, we also wish to show some elemetary meas i creatig the quatitative formulas cocerig the rate of covergece of a sequece. This article cotais also the message that mathematics computers do ot exclude each other, quite the cotrary, they complemet oe aother. This is illustrated by the fact that Mathematica Maple softwares have the Euler Maclauri formula built-i. This is the reaso why these programs execute some umerical summatios so fast.

4 Wallis sequece estimated through the Euler MacLauri formula 33 Trasformig the Wallis product By puttig a suitable weights o the terms of Wallis sequece, usig the Euler Maclauri formula, we ca obtai a faster coverget sequece. By a, due to the cotiuity of the logarithmic fuctio, we have But, accordig to, where l W = lim l W = l W. 3 4k l 4k fk, 4 4x fx l 4x l + 4x l 4 + l x l 4x, Fuctio f has the derivatives f x 4x 3 x, f x x 4x 3 x, 5c f 3 x 384x4 48x + 4 x x 3 x x 3 x 3 < 0, 5d for x. Because all derivatives of fuctio fx coverge to 0 as x teds towards ifiity we ca use the Euler Maclauri summatio formula of order, or 3 for the sum S, S := fk 6 figurig i 4. For example, i [7, p. 8, item 3a, there is stated that S = Sm + where, due to [7, p. 7, item b, I 7a we eed also beig the primitive of f. From 4 7c we fid [fm + f + [f f m + ρ 3 m, = 6 F x l l W = S = Sm fm + f hece where m m P 3 x f 3 x dx. fx dx + ρ 3 m,, 5a 5b 7a 7b x 4x x x + 4x, 7c [ l W = Cm + f + f + F Cm Sm fm + [f f m + F F m + ρ 3 m,, f m + ρ 3 m,, 8 F m

5 33 Vito Lampret depeds oly o m. Accordig to 5a, 5b, 7c, 7b the equality 8 ca be expressed i the form l W = Cm + 4 l l + 4 P 3 x f 3 x dx 6 for > m. As grows beyod all limits expressio l W approaches to l W, cosiderig a. Further, 4 lim l + 4 = 0 because 4 4 [ + m coverges towards e 0 = as teds to ifiity. Moreover, by [7, p. 5, item 8a, Beroulli periodic fuctio P 3 x is bouded P 3 x <, x R. 0 0 Thus, the itegral m P 3 x f 3 xdx is absolutely coverget due to 5d 5c. Hece, lettig to grow beyod all limits i 9, we obtai l W = Cm 6 m P 3 x f 3 x dx. Subtractig equatio 9 from equatio we fid the expressio l W 4 W = l l P 3 x f 3 x dx. 6 }{{} =δ a By 5d derivative f 3 does ot chage its sig. Cosequetly, by the mea value theorem because of the periodicity of Beroulli fuctio P 3, there exists, for each iteger, some ξ [0,, such that P 3 x f 3 x dx = P 3 ξ f 3 x dx = P 3 ξ f f b [ = P 3 ξ 4 3. Therefore, the remaider i a ca be estimated as δ 6 δ := 6 P 3 x f 3 x dx < 80.

6 Wallis sequece estimated through the Euler MacLauri formula 333 Hece, by a b, there exists, for each iteger, some ϑ [,, such that W = W + 4 exp ϑ 80 3a cosequetly W = W exp 6 4 ϑ 80 3b 3 Approximatig π. Sice W = π, by a, we est π, accordig to 3a, as π < π < π, valid for ay iteger, where π :=W + 4 exp a 4b π :=π exp 40. 4c Figure illustrates the covergece of sequeces π π Π Figure. Covergece of sequeces π π. Accordig to, the Wallis sequece W is icreasig mootoically coverges towards π. Thus, for ay positive iteger we have W < π. Cosequetly, from 4b 4c, we obtai the estimates π π = W + [ 4 exp exp 40 < π + e exp , that is π π < 3 0, 4d

7 334 Vito Lampret for every iteger. Here we used the simple iequality e x < +e x, valid for 0 < x. Example. By 4d we estimate π 000 π 000 < so we expect to obtai twelve decimal places of π by puttig = 000 ito 4a 4c. Direct computatio gives π 000 = π 000 = , i.e. π is really determied to twelve decimal places as π = Computig the Wallis products. Similarly, as we derived 4a, we obtai from 3b the estimate where W := π W := π W < W < W, 4 exp 4 exp a 5b 5c for ay iteger. From these relatios we ca compute W for rather large. However, for a very large a direct computatio usig these formulas is ot easy due to the third factor i 5b 5c, which varies like exp, accordig to the well kow 4 covergece towards the expoet fuctio. The rate of this covergece is evidet from the followig lemma: Lemma For ay positive real x t x there holds the estimate exp x x < + x t < exp x x. t t 3t Ideed, itegratig the iequality valid for t 0,, we obtai the estimate y y < t < + t < 3 t y 0 dt + t = l + y < y y 3 valid for y 0,. Because for ay x > 0 t x the umber y := x t lies i the iterval 0,, we ca put this y ito the relatios above to obtai x t x t < l + x < x t t x 3t, which verifies the propositio. Accordig to the just verified lemma, where we put t = x = 4, we fid that exp 4 < < exp for every iteger. From 5a 6 we deduce the estimate W < W < W, 7a

8 where Wallis sequece estimated through the Euler MacLauri formula 335 W := π W := π + + exp ϕ, exp ψ, 7b 7c ϕ := , 7d ψ = e for ay iteger. For such we have < ϕ < ψ < 7f 4 4 for 8 there holds the estimate < ϕ < ψ <. 7g 4 4 Ideed, from 7d there follows [ ϕ = [ [ > = + 4 λ }{{} =λ from 7e we fid ψ < < 4 + [ [ < }{{} =µ = 4 µ. Factor λ icreases µ decreases, hece λ λ > 0.9 µ µ <.0 for, moreover λ λ8 > µ µ8 <.0000 for 8. This cofirms 7f 7g. I order to simplify 7a 7e, we use the Taylor s formula of order for the expoetial fuctio, to produce the estimate + x < e x < + + ex x x, 8a valid for positive x. Thus, by 7f, we have exp ϕ > + ϕ exp ψ < + ωψ ψ, 8b 8c

9 336 Vito Lampret where ωx := + e x x/ icreases for x > 0. Thus, due to 7f 7g, there holds the estimate.0.0 ωψ < ω ω < for.0000 ωψ < ω ω 4 for 8. Hece, accordig to 8c, we obtai for exp ψ < ψ.000 < exp ψ < ψ for 8. Cosiderig 7a 7c 8b, 8c we fid where, for, a := π a < W < b, b := π + but for 8 we ca take more accurate boud b := π 8d 8e 9a + ϕ 9b ψ, 9c +.007ψ. 9d Figure shows the graph of sequece b a for Figure. Graph of sequece b a. 5 Estimatig the rate of covergece of the Wallis products By 9a 9d 7f 7g we obtai a > π + + δ + 4 b < π = π δ δ = π δ, + 4 +

10 Wallis sequece estimated through the Euler MacLauri formula 337 where we ca take δ = 0. δ = 0. for similarly δ = 0.00 δ = for 8. Cosequetly a > π. > π. + 4 for, a > π.00 + for 8. Similarly, for, we have b < π for 8 we obtai b < π > π.00 4 π < {}}{ π < {}}{ 50 Therefore, accordig to 9a, usig also direct computatios, we fid π. < W < π a for 3, π.00 < W < π b 4 4 for 6. This way we have estimated the rate of liear covergece lim W = π : π < π.00 W < π 0c 8 for every iteger 6. Cosequetly, the estimate 0.3 < π 0.4 W < 0d is true eve for 3, by usig direct computatio [6. The bouds could certaily be improved simply by usig the Euler Maclauri formula of order higher tha 3. I the literature better bouds are also kow, see for example [9 [5. Example. Accordig to 0c we have π < W < π Cosiderig 4d, we see that the covergece of sequece W towards π is much slower tha the covergece of sequeces π π. Modified Wallis sequeces π π have cosiderably accelerated covergece, relatively to the covergece of the origial Wallis sequece. Figure 3 illustrates relatio 0a by showig the graph of sequece π 0.8. W the graphs of fuctios π 8, π 8 for Remarks. R. If we should approximate the Wallis product W = + k by usig the Euler Maclauri summatio formula of order 3 for the fuctio x gx l l + l x l x, x

11 338 Vito Lampret Figure 3. Lower upper bouds π π W. π. 8 for the ull-sequece we should obtai the expressio π = W + e exp 6 exp ϑ 60 for some ϑ,. This formula is a little bit better tha c, but less accurate tha 3a. R. For the fuctios fx gx from 5a we should use the Euler Maclauri formula of the higher order tha the order three to obtai more accurate formula for π W. However, such formulas should be more complicated tha the oe just preseted. R3. I 665 3, te years after Wallis, Newto made the very first applicatio of his ew calculus. He computed π to sixtee decimal places usig the biomial expasio x [ dt x arcsi x = = t k x dt = k t k dt t k k 0 0 k=0 = x + 3 x x x7 +, where he put x = to obtai the expasio for π 6. Later, i 6664 or 67 5 or 680 6, he foud aother way to compute sixtee correct digits of π, usig his formula 7 employig terms of a ifiite series expasio: π = = /4 0 x x dx k= After a 5 digits computatio, durig the plague i Cambridge Lodo betwee Newto wrote: I am ashamed to tell you to how may figures I carried these computatios, havig o other busiess at the time [5. Perhaps Newto should have less 3 http: //wvwv.essortmet.com/pimathematicsa_rjar.htm; author Greg Fee 7 See [5, pp. 0& or [0, pp

12 Wallis sequece estimated through the Euler MacLauri formula 339 work i his computatio if the Euler Maclauri method of summatio should be kow to him. However this method has bee discovered about 60 years later ufortuately ot i its correct form with a remaider, which has bee preseted the first time by Poisso i 83 [. R4. Kowig the Euler Maclauri formula it is ot difficult to fid umerical sum of slowly coverget Leibiz series k+ k = π 4, which, like the Wallis product, also is cosidered i the literature as usuitable for umerical computatio of π. Refereces [ M. Abramowitz I. A. Stegu Hbook of Mathematical Fuctios, 9th ed. Dover Publicatios New York 974. [ G. B. Arfke H. J. Weber, Mathematical Methods for Physicists Harcourt/Academic Press 00. [3 D. H. Bailey, J. M. Borwei P. B. Borwei, Ramauja, Modular Equatios, Approximatios to Pi or How to compute Oe Billio Digits of Pi, Amer. Math. Mothly , 0 9. [4 P. Beckma, A History of π, Golem Press Boulder Colorado 970, 97, 977, 98 Hippocree Books 990 Marboro Books 990 St. Marti s Press 97, 976. [5 L. Berggre, J. Borwei P. Borwei, Pi: A Source Book Spriger Verlag New York 997. [6 D. Blater, The Joy of π Walker & Co [7 T. J. I. A. Bromwich, A Itroductio to the Theory of Ifiite Series, 3rd ed. Chelsea Publishig Compay New York 99. [8 D. Castellaos, The Ubiquitous Pi, Math. Mag , [9 J.T. Chu, A Modified Wallis Product Some Applicatios, Amer. Math. Mothly 69 96, [0 W. Duham, Jourey Through Geius: The Great Theorems of Mathematics Pegui Books 99. [ S.R. Fich, Mathematical Costats Cambridge Uiversity Press 003. [ R. L. Graham, D. E. Kuth O. Patashik, Cocrete Mathematics, Readig [etc. Addiso-Wesley 989. [3 G. H. Hardy, Diverget Series Claredo Press Oxford 956. [4 P. Herici, Applied Computatioal Complex Aalysis, Vol., 7th ed. Joh Wiley & Sos Ic. 99. [5 D. K. Kazarioff, O Wallis formula, Ediburgh Math. Notes , 9. [6 K. Kopp, Theory Applicatios of Ifiite Series Hafer New York 97. [7 V. Lampret, The Euler Maclauri Taylor Formulas: Twi, Elemetary Derivatios, Math. Mag , 09. [8 V. Lampret, A Ivitatio to Hermite s Itegratio Summatio: A Compariso betwee Hermite s Simpso s Rules, SIAM Rev , [9 J. Lewi M. Lewi, A Itroductio to Mathematical Aalysis, d ed. McGraw-Hill Ic [0 T. J. Osler, The Uio of Vieta s Wallis Products for Pi, Amer. Math. Mothly , [ S. D. Poisso, Mémoire sur le calcul umérique des itégrales défiies, Mémoires de l Académie Royale des Scieces de l Istitut de Frace, séries, 6 83, [ A. Sofo, Some Represetatios of Pi, AustMS Gazette 3 004, [3 H. C. Schepler, The Chroology of Pi, Math. Mag , 65 70, 6 8, [4 J. Wallis, Computatio of π by Successive Iterpolatios, 655 i: A Source Book i Mathematics, D.J. Struik, Ed. Harvard Uiversity Press Cambridge MA 969, [5 E.T. Whittaker G.N. Watso, A Course of Moder Aalysis Cambridge Uiversity Press 95, 7. [6 S. Wolfram, Mathematica, versio 5.0 Wolfram Research Ic., Uiversity of Ljubljaa, Ljubljaa, Sloveia vlampret@fgg.ui-lj.si Received 0 Jue 004, accepted 8 September 004.