Even from Gregory-Leibniz series π could be computed: an example of how convergence of series can be accelerated

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1 Lecturas Matemáticas Volumen 27 (2006), páginas 2 25 Even from Gregory-Leibniz series could be computed: an example of how convergence of series can be accelerated Vito Lampret University of Ljubljana, Ljubljana, Slovenia Abstract. Gregory-Leibniz series Σ j= ( )j =,isdiscussedin 2j 4 literature as an example of beautiful, interesting and simple analytic expression for. Unfortunately, this series is considered by most authors as unsuitable for computation of. It is shown that, using a simple transformation of the series and applying the Euler Maclaurin summation formula, slow convergence can be accelerated to the point where numerical computation of can be performed accurately to several decimal places. It is also suggested that the Euler Maclaurin formula, of some low order, should be included in the undergraduate mathematics curriculum. Key words and phrases. acceleration, alternating, convergence, estimate, Euler-Maclaurin, Gregory-Leibniz, remainder, series AMS Mathematics Subject Classification. 40A05, 40A25, 65B0, 65B5, 97D30, 97D40. Resumen. La serie de Gregory-Leibniz Σ j= ( )j =,seconoce 2j 4 en la literatura como un ejemplo de una hermosa, interesante y sencilla expresión analítica de. Desafortunadamente, la mayoría de los autores la consideran inapropiada para el cálculo de. En esta nota se usa una simple transformación de la serie y una aplicación de la fórmula sumación de Euler-Maclaurin para mostrar cómo esta convergencia lenta puede acelerarse hasta el punto de que el cálculo numérico de nos produzca un valor bastante preciso con varias cifras decimales. También se sugiere que la fórmula de Euler-Maclaurin debiera incluirse en los cursos de matemáticas del pregrado. Since next year is the third-centenary of Euler s birth, April 5, 707, we dedicate this contribution to him, the main creator of the famous Euler-Maclaurin summation formula.

2 22 Vito Lampret It seems that the Gregory-Leibniz series, figuring in the identities () 4 = j= ( ) j 2j = k= ( ) k 2k, is not suitable for computation of. estimate [6, p ] Indeed, from the well known Leibniz 0 a n a n2 ( ) n r n a n of remainder r n := k=n a k of alternating series k= ( )k a k, where a n monotonically decreasingly converges to zero, we obtain, for the remainder in series (), the estimate (a) 2 (2n )(2n 3) ( )n r n 2n, valid for every positive integer n. Unfortunately, this estimate indicates slow convergence of series (). To compute from (), using (a), to six decimal places for example, we need to sum up much more than five millions terms of series () due to rounding the terms of series during the calculations. Figure illustrates relation (a) by showing the graphs of sequences n ( ) n r n 4 s n and n /(2n ) for series () Figure : Graphs of sequences n ( ) n r n and n /(2n ). In order to accelerate convergence in () we first substitute alternating conditionally convergent series () by absolutely convergent one. Indeed, in partial sums of () with even indexes, we collect pairwise the terms with odd and even This estimate is sharp as it could be seen from the example of geometric series.

3 Even from Gregory-Leibniz series could be computed 23 indexes 2n k= ( ) k 2k = = n n n 4i 4i 2 6i 2. This way, using (), we get the expression (2) 4 = 2 6i 2. The obtained series converges faster than the original one. To accelerate also convergence in (2) we use the Euler-Maclaurin summation formula of order four [3, p. 320, Theorem ] or [4, p. 9, Theorem 2]. It says that for any m N and any function f C 4 [, ), such that f (4) (x) does not change sign and lim n f(n) = lim n f (n) = lim n f (3) (n) =0and f (4) (x) dx converges, there exists a number ω(m), depending on m, such that (3) and m f(k) = f(k) k= k= m f(x) dx f(m) 2 (3a) 0 ω(m) 384. f (m) 2 ω(m) f (3) (m) The infinite series in (2) originates from function ϕ :[, ) R, such that (4) (4a) (4b) (4c) (4d) Because ϕ(x) 6x 2 ϕ 32x (x) (6x 2 ) 2 < 0 ϕ (2) (x) 32 ( 48x 2 ) (6x 2 ) 3 > 0 ϕ (3) (x) 644 ( 6x 3 x ) (6x 2 ) 4 < 0 ϕ (4) (x) 644 ( 280x 4 60x 2 ) m (6x 2 ) 5 > 0. ϕ(x) dx = ( ) 4m 8 ln, 4m

4 24 Vito Lampret we obtain from equations (3) (3a) and (4) (4c), for any m N, the following expression m 6i 2 = 6i 2 ( ) 4m 8 ln 4m 2(6m 2 ) ( 8m 644 6m 3 m ) 3(6m 2 2 ω(m) ) (6m 2 ) 4. Hence, from (2) we conclude (5) = m ρ m, where [ (5a) m =4 8 m 6i 2 ln and (5b) 0 ρ m 8ω(m) 644 ( 6m 3 m ) for every m N. (2m ) 5. (6m 2 ) 4 < ( ) 4m 4m 4 6m 2 ] 64m 3(6m 2 ) 2 32 (4m )(4m ) 4 < (2m ) 5 Figure 2 shows graphs of sequences m ρ m and m Figure 2: Graphs of sequences m ρ m and m (2m ) 5. Using [7] we find from (5a) that 0 = and from (5b) that 0 ρ 0 < Hence < = This way we determined to six decimal places = We point out that during this computation we have to sum only 3 terms. To obtain more decimal places of we should increase parameter m. However, sequence m m converges relatively slowly and therefore is not suitable for the computation of large number of digits of. For example, we have 000 > and ρ 000 < Thus to obtain 7 decimal places we should sum much more than 000 terms, due to their rounding during

5 Even from Gregory-Leibniz series could be computed 25 the computation. Faster convergent sequence, originating in series (2), could be obtained using the Euler-Maclaurin formula of higher order than four [4]. References [] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 994. [2] K. Knopp, Theory and Applicattions of Infinite Series, Hafner Publishing Company, New York, 97. [3] V. Lampret, An Invitation to Hermite s Integration and Summation: A Comparison between Hermite s and Simpson s Rules, SIAM Rev. 46 (2) (2004), [4] V. Lampret, The Euler-Maclaurin and Taylor Formulas: Twin, Elementary Derivations, Math. Mag. 74 (200), [5] V. Lampret, Wallis sequence estimated through the Euler-Maclaurin formula: even from the Wallis product could be computed fairly accurately, AustMS Gazette 3 (5) (2004), [6] M. H. Protter and C. B. Morrey, A First Course in Real Analysis 2nd ed, Springer-Verlag N.Y., UTM 993. [7] S. Wolfram, Mathematica, version 5.0, Wolfram Research, Inc., (Recibido en septiembre de Aceptado para publicación en febrero de 2006) Vito Lampret Faculty of Civil and Geodetic Engineering University of Ljubljana Ljubljana 000, Slovenia vlampret@fgg.uni-lj.si, vito.lampret@fgg.uni-lj.si