A267315 - OEIS

%I #30 Sep 08 2022 08:46:15

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

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

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

%N Decimal expansion of the Dirichlet eta function at 4.

%H OEIS Wiki, <a href="https://oeis.org/wiki/Zeta_functions#Euler.27s_alternating_zeta_function">Euler's alternating zeta function</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletEtaFunction.html">Dirichlet Eta Function</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_eta_function">Dirichlet Eta Function</a>.

%F eta(4) = Sum_{k > 0} (-1)^(k+1)/k^4 = (7*Pi^4)/720.

%F eta(4) = Lim_{n -> infinity} A120296(n)/A334585(n) = (7/8)*A013662. - _Petros Hadjicostas_, May 07 2020

%e eta(4) = 1/1^4 - 1/2^4 + 1/3^4 - 1/4^4 + 1/5^4 - 1/6^4 + ... = 0.9470328294972459175765032344735219149279070829288860...

%t RealDigits[(7 Pi^4)/720, 10, 120][[1]]

%o (PARI) 7*Pi^4/720 \\ _Michel Marcus_, Feb 01 2016

%o (Magma) pi:= 7*Pi(RealField(110))^4 / 720; Reverse(Intseq(Floor(10^100*pi))); // _Vincenzo Librandi_, Feb 04 2016

%o (Sage) s = RLF(0); s

%o RealField(110)(s)

%o for i in range(1,10000): s += -((-1)^i/((i)^4))

%o print(s) # _Terry D. Grant_, Aug 04 2016

%Y Cf. A002162, A013662, A072691, A120296, A197070, A334585.

%K nonn,cons

%O 0,1

%A _Ilya Gutkovskiy_, Jan 13 2016