Ramanujan-type approximation series of the constant Ï and other classical ... but much earlier, the Indian mathematician Srinivasa Ramanujan [17] defined 17.
ERROR ESTIMATES OF RAMANUJAN-TYPE SERIES CRISTINEL MORTICI Abstract. The aim of this paper is to establish some inequalities related to Ramanujan-type approximation series of the constant and other classical constants such as Apéry’s constant and Catalan’s constant.
1. Introduction The number has fascinated mathematicians, and much e¤ort has been put into computing ever more of its digits and investigating its properties. Although for many purposes, scientists from other areas often use 3:14 or 22=7, in pure mathematics more precise approximations are necessary. Bailey, P. B. Borwein and Plou¤e [7] proposed the following fast series =
1 X
4 8k + 1
k=0
2 8k + 4
1 8k + 5
1 8k + 6
1 : 16k
(1.1)
that permits one to calculate the n-th hexadecimal or binary digit of computing any of the …rst n 1 digits, and the formula 1
9X = 8
1 X p = 3 k=0
16
2
8
6
2
2
1 3 1 + + 6k + 1 2 (6k + 2) 22 (6k + 3)
2
+
1 23 (6k + 5)
1
!
1 : 64k (6k + 1) (6k + 2) (6k + 3) (6k + 4) (6k + 5) k=0 (1.2) Since the publication of [7], other authors have presented similar formulas now known as BBP-type series. Chan [11] proposed the series 2
24
, without
2
1 ; 64k
(1.3)
but much earlier, the Indian mathematician Srinivasa Ramanujan [17] de…ned 17 famous series for 1= . We mention here 1
=
1
1 X ((2k)!) 42k + 5 6 16 4096k (k!) 3
(1.4)
k=0
1 X 27 (2k)! (3k)! 15k + 2 = 5 4 1458k (k!) k=0 p 1 5 15 X (6k)! 11k + 1 = 3 6 (3k)! (k!) 54000k k=0
1991 Mathematics Subject Classi…cation. 33B15; 26D15. Key words and phrases. inequlities; approximations, asymptotic formulas. 1
(1.5)
(1.6)
2
CRISTINEL M O RTICI
1
=
p
1
8 X (4k)! 1103 + 26390k : 4 9801 3964k (k!)
(1.7)
k=0
Series (1.7) produces to any new term an additional 8 correct digits in the result, and in 1985 R. W. Gosper used this formula to compute 17 million digits of : Since then, series of Ramanujan’s type were wide studied by other authors, and we refer here to [12] p 1 k 1500 3 X ( 1) (2k)! (3k)! 14151k + 827 (1.8) = 5 3003k (k!) k=0
and to the following series [13] p 1 k X 6403203 ( 1) (6k)! 545140134n + 13591409 ; = 3 12 6403203k (3k)! (k!)
(1.9)
k=0
where each term contributes 14 digits of : Excellent surveys on Ramanujan’s series are [4], or [10]. We establish in this paper some inequalities for evaluating the error estimate for this type of approximation series. It is to be noticed that knowledge of these error estimates can also be useful in the problem of determining the irrationality, or transcendentally of the involved constants. In this sense, remember that the proof of irrationality of e is based on the fast approximation 1 1 1 e 1+ + + + ; 1! 2! n! where the error estimate is n =n!n; with 0 < n < 1: 2. The Results In order to illustrate our method, we …nd the error estimates committed to the following approximations related to (1.1), (1.2), and (1.4) n X 1 16k
k=0 n
2
9X 8
k=0
16 (6k + 1)
2 8k + 4
4 8k + 1 24
2
8
(6k + 2) 1
1 8k + 5
2
n
(6k + 3)
1 8k + 6 6
2
(6k + 4)
1 X ((2k)!) 42k + 5 := 6 16 4096k (k!)
2
+
:=
n
1 (6k + 5)
2
!
1 := 64k
3
n;
k=0
but we can similarly proceed to obtain the error estimates for the other series (1.1)(1.9), or others approximating classical constants, such as Apéry’s constant and Catalan’s constant. With the notations 2 1 1 4 z (k) = 8k + 1 8k + 4 8k + 5 8k + 6 24 8 6 1 16 t (k) = 2 2 2 2 + 2 (6k + 1) (6k + 2) (6k + 3) (6k + 4) (6k + 5) r (k) = 42k + 5; we give the following theorems.
