A092731 Decimal expansion of Pi^5.
3, 0, 6, 0, 1, 9, 6, 8, 4, 7, 8, 5, 2, 8, 1, 4, 5, 3, 2, 6, 2, 7, 4, 1, 3, 1, 0, 0, 4, 3, 4, 3, 5, 6, 0, 6, 4, 8, 0, 3, 0, 0, 7, 0, 6, 6, 2, 8, 0, 7, 4, 9, 9, 0, 5, 5, 3, 4, 9, 2, 4, 4, 3, 6, 8, 6, 2, 3, 4, 9, 9, 2, 1, 3, 3, 6, 1, 4, 0, 2, 4, 4, 8, 5, 7, 8, 3, 5, 0, 0, 4, 7, 3, 5, 0, 5, 1, 1, 8, 9, 0, 4, 0, 3, 7
Offset: 3
Examples
306.0196847852814532
Links
- G. C. Greubel, Table of n, a(n) for n = 3..10000
- Jean-Christophe Pain, The fifth power of Pi: new series representation involving the golden ratio and an application in physics, 2022.
- Jean-Christophe Pain, New series representations for any positive power of Pi from a relation involving trigonometric functions, 2208.02624 [math.NT], 2022.
- Kh. Hessami Pilehrood and Tatiana Hessami Pilehrood, Series acceleration formulas for beta values, Discr. Math. Theor. Comp. Sci. 12 (2) (2010) 223-236.
- Index entries for transcendental numbers
Programs
-
Magma
R:= RealField(100); (Pi(R))^5; // G. C. Greubel, Mar 09 2018
-
Mathematica
RealDigits[Pi^5, 10, 100][[1]] (* G. C. Greubel, Mar 09 2018 *)
-
PARI
Pi^5 \\ G. C. Greubel, Mar 09 2018
Formula
From Peter Bala, Oct 31 2019: (Start)
Pi^5 = (4!/(2*305)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/6)^5 + 1/(n + 5/6)^5 ), where 305 = ((3^5 + 1)/4)*A000364(2) = A002437(2).
Pi^5 = (4!/(2*3905)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/10)^5 - 1/(n + 3/10)^5 - 1/(n + 7/10)^5 + 1/(n + 9/10)^5 ), where 3905 = ((5^5 - 1)/4)*A000364(2).