A142970 Wolfdieter Lang (after loss rewritten, Aug 24 2019) A continued fraction for Pi - 3 = A000796 - 3 is C = b(0) + K(a(n)/b(n)) with b(0) = 0, b(n) = 6, a(n) = (2*n - 1)^2, for n >= 1. See the Delahaye reference, and the Berndt et al. link, Entry 25, p. 268, with references to Euler, Stieltjes and Perron. [In this link, if the given expression is called R(n, x) then Pi = 4/(R(0,3)]. The n-th approximant is the rational r(n) = A(n)/B(n), with A(n) = A142970(n), with A(-1) = 1, and B(n) = A001879(n) = (n + 1)*(2*n + 1)!!, with B(-1) = 0. These approximants are not always in lowest terms. The numerators A(n) = A142970(n), for n=0..25, are: [0, 1, 6, 61, 660, 8901, 133266, 2303865, 43808040, 928665225, 21386693790, 537861526965, 14540730176700, 423407835413325, 13140639311294250, 434929825450371825, 15237733330856005200, 565064979900590948625, 22056613209702152061750, 905913636742121921038125, 38983590512409704812150500, 1756742366437965178137991125, 82621113056073335266494221250, 4053129970373319497328397355625, 206828818563105914587656118875000, 10972537970244975600611418764105625, ...]. The denominators B(n) = A001879(n), for n=0..25, are: [1, 6, 45, 420, 4725, 62370, 945945, 16216200, 310134825, 6547290750, 151242416325, 3794809718700, 102776096548125, 2988412653476250, 92854250304440625, 3070380543400170000, 107655217802968460625, 3989575718580595893750, 155815096120119939628125, 6396619735457555416312500, 275374479611447760672253125, 12404964652972837218854831250, 583597200719403932796125015625, 28621636626586418964957783375000, 1460896036148681801336386859765625, 77485925757326082742881959041968750, ...]. The rationals r(n),for n = 0..25, are in lowest terms : [0/1, 1/6, 2/15, 61/420, 44/315, 989/6930, 6346/45045, 51197/360360, 36056/255255, 4127401/29099070, 2057402/14549535, 189721879/1338557220, 236723324/1673196525, 1422382919/10039179150, 20600649518/145568097675, 10227626700773/72201776446800, 638723926928/4512611027925, 1278290544991/9025222055850, 23635180313246/166966608033225, 94585786464329/667866432132900, 969106771716436/6845630929362225, 83372817133541471/588724259925151350, 41673480936996358/294362129962575675, 15673494950136175183/110680160865928453800, 13711014028962429224/96845140757687397075, 27427870902012803389/193690281515374794150, ...]. -------------------------------------------------------------------------------------------------- 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). In the present case one finds, with the given values for B(j), c(0) = 0, c(j) = (-1)^(j-1)/(j*(j+1)*(2*j+1)). Therefore, C = sum(((-1)^(j-1))/(j*(j+1)*(2*j+1)), j=0..infinity). --------------------------------------------------------------------------------------------------- The approximants r(n), for n = 10^k, with k=0..3, are (Maple 20 digits): .16666666666666666667, .14140671849650177824, .14159241097198067426, .14159265334054205190. This should be compared with the value of Pi - 3 = A000796 - 3 which is approximately (Maple 20 digits) .1415926535897932385. ---------------------------------------------------------------------------------------------------- The standard regular continued fraction for Pi - 3 is [0, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2,...] with approximants {R(n) = N(n)/D(n)} given (in lowest terms) by N = {0, 1, 15, 16, 4687, 4703, 9390, 14093, 37576, 51669, 192583, 244252, 3612111, 7468474, 11080585, 18549059, 48178703, 114906465, 277991633, 670889731,...}, and D = {1, 7, 106, 113, 33102, 33215, 66317, 99532, 265381, 364913, 1360120, 1725033, 25510582, 52746197, 78256779, 131002976, 340262731, 811528438, 1963319607, 4738167652,...}. R(10^2) is approximately .14159265358979323846 (Maple 20 digits). to be compared with the above approximation .14159241097198067426 which is not so close to Pi - 3. --------------------------------------------------------------------------------------------------------- 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 ----------------------------------------------------------