%I #53 Nov 06 2023 02:03:11

%S 9,7,4,0,9,0,9,1,0,3,4,0,0,2,4,3,7,2,3,6,4,4,0,3,3,2,6,8,8,7,0,5,1,1,

%T 1,2,4,9,7,2,7,5,8,5,6,7,2,6,8,5,4,2,1,6,9,1,4,6,7,8,5,9,3,8,9,9,7,0,

%U 8,5,5,4,5,6,8,2,7,1,9,6,1,9,0,1,2,1,8,6,7,2,3,4,7,5,2,9,9,2,5,5

%N Decimal expansion of Pi^4.

%H Harry J. Smith, <a href="/A092425/b092425.txt">Table of n, a(n) for n = 2..20000</a>

%H Mohammad Reza Yegan, <a href="https://doi.org/10.1016/j.jnt.2017.02.009">On the irrationality of Pi^4 and Pi^6</a>, Journal of Number Theory, Volume 178, September 2017, Pages 5-10.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals 120 * Sum_{j>=1} Sum_{i=1..j-1} 1/(i*j)^2. - _Enrique Pérez Herrero_, Jun 29 2012

%F Equals Sum_{k>=1} k*(k+1)*(k+2)*zeta(k+3)/2^(k-1). - _Amiram Eldar_, May 21 2021

%F From _Peter Bala_, Oct 21 2023: (Start)

%F Pi^4 = 90*Sum_{n >= 1} 1/n^4 (Euler).

%F The following faster converging series representations for the constant Pi^4 may be easily verified using partial fraction expansions of the summands of the series. Presumably, these are the first three cases of an infinite family of similar results.

%F Let P(n) = n*(n + 1)*(n + 2)/2!. Then Pi^4 = 1575/16 - 15*Sum_{n >= 1} d/dn(P(n))/P(n)^4.

%F Let Q(n) = n*(n + 1)*(n + 2)*(n + 3)*(n + 4)/4!. Then Pi^4 = 673165/6912 + Sum_{n >= 1} d/dn(Q(n))/Q(n)^4.

%F Let R(n) = n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)/6!. Then Pi^4 = 5610787/57600 - (3/56)*Sum_{n >= 1} d/dn(R(n))/R(n)^4.

%F Taking 10 terms of the last series gives the approximation Pi^4 = 97.4090910340

%F 024372(50...), correct to 16 decimal places. (End)

%e 97.40909103400243723644033268870511124972758567268542169146785938997085...

%t RealDigits[Pi^4, 10, 100][[1]] (* _G. C. Greubel_, Mar 09 2018 *)

%o (PARI) default(realprecision, 20080); x=Pi^4/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b092425.txt", n, " ", d)); \\ _Harry J. Smith_, Jun 22 2009

%o (Magma) R:= RealField(150); (Pi(R))^4; // _G. C. Greubel_, Mar 09 2018

%o (Magma) R:=RealField(110); SetDefaultRealField(R); n:=Pi(R)^4; Reverse(Intseq(Floor(10^98*n))); // _Bruno Berselli_, Mar 12 2018

%Y Cf. A000796 (Pi), A002388 (Pi^2), A091925 (Pi^3), A092731 (Pi^5), A092732 (Pi^6), A092735 (Pi^7), A092736 (Pi^8), A058286 (continued fraction), A013662.

%K cons,easy,nonn

%O 2,1

%A _Mohammad K. Azarian_, Mar 22 2004