OFFSET
0,2
COMMENTS
This Madhava series results from the arctan series for tan(Pi/6) = sqrt(3)/3 = A020760.
REFERENCES
L. B. W. Jolley, Summation of Series, Dover (1961), eq. (273), pp. 16 and 17.
Ian Stewart, Grössen der Mathematik, Rowohlt Tachenbuch Verlag, Nr. 63394, 2020, p. 74. [English Original: Significant Figures. Lives and Works of Traiblazing Mathematicians, Profile Books, London, 2017]
FORMULA
a(n) = numerator(Sum_{j=0..n} (-1)^j/((2*j+1)*3^j)), for n >= 0.
EXAMPLE
The partial sums begin: 1/1, 8/9, 41/45, 856/945, 23147/25515, 254512/280665, 3309041/3648645, 29780368/32837805, 168757087/186080895, ...
For n = 100 the partial sum is 0.9068996821171089252970391288210778661420331240463726... compared to 0.9068996821171089252970391288210778661420331240463702...(the first 53 digits coincide).
MATHEMATICA
Numerator @ Accumulate @ Table[(-1)^j/((2*j + 1)*3^j), {j, 0, 20}] (* Amiram Eldar, Apr 08 2022 *)
PROG
(PARI) a(n) = numerator(sum(j=0, n, (-1)^j/((2*j+1)*3^j))); \\ Michel Marcus, Apr 08 2022
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Apr 07 2022
STATUS
approved