A352397 Numerators of partial sums of the Madhava series for Pi/(2*sqrt(3)) = A093766.

1, 8, 41, 856, 23147, 254512, 3309041, 29780368, 168757087, 28857376792, 259716622073, 5973480691064, 89602217802389, 7257779456082784, 210475605899597261, 6524743766713282016, 19574231315333822573, 6524743770186190936, 2172739675639135323463, 19554657080276529569192

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]

Links

Table of n, a(n) for n=0..19.

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

Cf. A352398 (denominators).

Cf. A020760, A093766.

Sequence in context: A228480 A067301 A295936 * A204572 A086392 A234860

Adjacent sequences: A352394 A352395 A352396 * A352398 A352399 A352400

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Apr 07 2022

Status

approved