OFFSET
1,2
COMMENTS
Consider infinite sum made of areas of circles Pi*radius^2 with diameter 1/n.
The volume is (Pi/4)*(1 + 1/4 + 1/9 + 1/16 + 1/25 + ... + 1/n^2)
= (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...)*(1 + 1/4 + 1/9 + 1/16 + 1/25 + ...)
= (Pi/4) * (Pi^2/6) = Pi^3/24.
Equals volume of a cone of height Pi^2/8 and radius 1.
Equals volume of a sphere (4*Pi*Pi^2/32)/3 with radius^3 = (Pi^2/32).
LINKS
FORMULA
Equals Integral_{x=0..oo} arctan(x)^2/(x^2 + 1) dx. - Amiram Eldar, Aug 06 2020
EXAMPLE
1.291928195012492507311513127795891466759387023578...
MATHEMATICA
RealDigits[(Pi^3)/24, 10, 50][[1]] (* G. C. Greubel, Jun 09 2017 *)
PROG
(PARI) (Pi^3)/24 \\ G. C. Greubel, Jun 09 2017
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Eric Desbiaux, Dec 08 2008
STATUS
approved