is Weisstein's name for the x = pi case of inf /===\ | | x | | cos(-), | | n n = 3 the inner/outer radius ratio of the annular nesting of all the regular polygons, alternated with circles. (With the right z(r), might make a nice drinking cup.) Perhaps equivalent to his Mathworld circumscribing formula, simply expand the log of the quadrisected product, inf inf inf /===\ /===\ /===\ | | x | | x | | x log(( | | cos(-------)) ( | | cos(-------)) ( | | cos(-------)) | | 4 n + 3 | | 4 n + 4 | | 4 n + 5 n = 0 n = 0 n = 0 inf /===\ | | x | | cos(-------)) | | 4 n + 6 n = 0 4 i 2 i 2 B pi inf 2 i 2 i i 2 i 2 i ==== (1 - 2 ) B (--------------- + (- 1) (2 + 1)) x \ 2 i 2 (2 i)! = > ---------------------------------------------------------- , / 2 i (2 i)! ==== i = 1 a series with convergence radius 3 pi/2, hence 2 lg 3 - 2 ~ 1.17 bits/term. Note the unusual Bern^2. Because B_n ~ n!/(2 pi)^n, the radius appears to be only pi/2 (useless), but the factor 4 i 2 i 2 B pi i 2 i 2 i i 2 i (- 1) 2 --------------- + (- 1) (2 + 1) ~ - ----------- 2 (2 i)! 2 i 3 saves the day, at the cost of 3.17 bits/term of subtractive cancellation. Adding more terms without raising the intermediate precision will actually ruin your accuracy. To 30 places, 1/8.70003662520819450322240985911. --rwg