A142969 Wolfdieter Lang (after loss rewritten, Aug 25 2019) A continued fraction for 4/Pi - 1 = A088538 - 1 is C = b(0) + K(a(n)/b(n)) with b(0) = 0, b(n) = 2, a(n) = (2*n - 1)^2, for n >= 1. This is Brouncker's continued fraction for 4/Pi without the leading 1. See, e.g., the Brezinski reference, pp. 78-79. The n-th approximant is the rational r(n) = A(n)/B(n) with the standard recurrences q(n) = b(n)*q(n-1) = a(n)*q(n-2) for q from A or B, with with A(-1) = 1, A(0) = 0 and B(-1) = 0, B(0) = 1. These rationals {r(n)} are not always in lowest terms. The numerators A(n) = A024200(n+1) are for n = 0..20: [0, 1, 2, 29, 156, 2661, 24198, 498105, 6440760, 156833865, 2638782090, 74441298645, 1544798322900, 49615408298925, 1225388793991950, 44177335967379825, 1265953302961023600, 50641025474398676625, 1652074847076051263250, 72631713568603890826125, 2658069269539881753055500, ...]. The denominators B(n) = A024199(n+1) = (2*n+1!!*sum((-1)^k/(2*k+1), k=0..n) are for n = 0..20: [1, 2, 13, 76, 789, 7734, 110937, 1528920, 28018665, 497895210, 11110528485, 241792844580, 6361055257725, 163842638377950, 4964894559637425, 147721447995130800, 5066706567801827025, 171002070002301095250, 6548719685561840296125, 247199273204273879989500, 10455001188148106850385125, ...]. The rationals r(n), now in lowest terms, are for n = 0..25: [0, 1/2, 2/13, 29/76, 52/263, 887/2578, 8066/36979, 11069/33976, 143128/622637, 3485197/11064338, 2792362/11757173, 78773861/255865444, 326941444/1346255081, 1166735057/3852854518, 28815727078/116752370597, 1038855637093/3473755390832, 902109848368/3610501179557, 1031041592023/3481569435902, 33635927876926/133330680156299, 37917122954701/129049485078524, 1387635433109516/5457995496252709, 66513954553071413/227848175409504262, 59972573887236398/234389556075339277, 3113073102662686381/10721947005578370344, 19815080274604367992/77030060483083029083, 21714004603186473817/75131136154500923258, ...]. In lowest terms one has the numerators N(n) = A142969(n) and the denominators D(n) = A007509(n), for n >= 0 (with A142969(0) = 0). N = [0, 1, 2, 29, 52, 887, 8066, 11069, 143128, 3485197, 2792362, 78773861, 326941444, 1166735057, 28815727078, 1038855637093, 902109848368, 1031041592023, 33635927876926, 37917122954701, 1387635433109516, 66513954553071413, 59972573887236398, 3113073102662686381, 19815080274604367992, 21714004603186473817, ...], D = [1, 2, 13, 76, 263, 2578, 36979, 33976, 622637, 11064338, 11757173, 255865444, 1346255081, 3852854518, 116752370597, 3473755390832, 3610501179557, 3481569435902, 133330680156299, 129049485078524, 5457995496252709, 227848175409504262, 234389556075339277, 10721947005578370344, 77030060483083029083, 75131136154500923258,...]. -------------------------------------------------------------------------------------------------- Euler [1] showed how to convert the continued fraction C = b(0) + K(a(n)/b(n)) to an infinite series. See the Brezinski reference [1], p. 98, or the Jones and Thron reference [3], section 2.3.1., pp. 36-38. The partial sums s(n) = sum(c(j),j=0..n) with c(0) = b(0), c(j) = (-1)^{j-1} product(a(k),k=1..j)/(B(j)*B(j-1)), for j >= 1, are then equal to the n-th approximant r(n) of C. In the present case one finds, with the above given values for B and a, c(0) = 0, c(j) = (-1)^(j-1))*((2*j-1)!!)^2/(B(j)*B(j-1)) = (-1)^(j-1))/((2*j+1)*S(j)*S(j-1)), whith the rationals S(j) := sum(((-1)^k))/(2*k-1),k=1..j). Therefore, C = sum(((-1)^(j-1))/((2*j+1)*R(j)*R(j-1)), j=0..infinity). --------------------------------------------------------------------------------------------------- The approximants r(n), for n = 10^k, with k=0..3, are (Maple 20 digits): .50000000000000000000, .23750284188214292671, .26923952899303435848, .27283479368898552763. This should be compared with the value of 4/Pi - 1 = A088538 - 1 which is approximately (Maple 20 digits) .27323954473516268615. ---------------------------------------------------------------------------------------------------- The standard regular continued fraction for 4/Pi - 1 is [0, 3, 1, 1, 1, 15, 2, 72, 1, 9, 1, 17, 1,...] (compare this with the one for Pi/4 given in A070989) with approximants R(n) = N(n)/D(n) given in lowest terms by R = {0/1, 1/3, 1/4, 2/7, 3/11, 47/172, 97/355, 7031/25732, 7128/26087, 71183/260515, 78311/286602, 1402470/5132749, 1480781/5419351, 4364032/15971451, 5844813/21390802, 33588097/122925461, 39432910/144316263, 73021007/267241724, 769642980/2816733503, 842663987/3083975227,...} N = {0, 1, 1, 2, 3, 47, 97, 7031, 7128, 71183, 78311, 1402470, 1480781, 4364032, 5844813, 33588097, 39432910, 73021007, 769642980, 842663987, 2454970954, 5752605895, 117507088854, 123259694749, 733805562599,...}, and D = {1, 3, 4, 7, 11, 172, 355, 25732, 26087, 260515, 286602, 5132749, 5419351, 15971451, 21390802, 122925461, 144316263, 267241724, 2816733503, 3083975227, 8984683957, 21053343141, 430051546777, 451104889918, 2685575996367,...}. R(10^2) is approximately 0.27323954473516268615 (Maple 20 digits) coinciding with the approximate (20 digits) value of 4/Pi - 1. This should be compared with the worse approximate value 0.26923952899303435848. --------------------------------------------------------------------------------------------------------- References: [1] Claude Brezinski, History of Continued Fractions and Padé Approximants, Springer, 1991. [2] L. Euler, Introductio in Analysin Infinitorum, Vol. 1, (1748), Ch. 18. [3] William B. Jones and W. J. Thron, Continued Fractions, Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Company, 1980. ----------------------------------------- eof ----------------------------------------------------------