cp's OEIS Frontend

A275703 Decimal expansion of the Dirichlet eta function at 6.

%I A275703 #42 Feb 21 2021 10:11:00
%S A275703 9,8,5,5,5,1,0,9,1,2,9,7,4,3,5,1,0,4,0,9,8,4,3,9,2,4,4,4,8,4,9,5,4,2,
%T A275703 6,1,4,0,4,8,8,5,6,9,3,4,6,9,3,2,6,8,8,8,0,3,4,8,3,3,3,9,3,2,5,4,1,9,
%U A275703 6,8,0,2,1,8,6,2,7,1,7,1,3,5,7,3,9,3,7,2,9,1,1,2,7,9,5,5,9,4,6,4
%N A275703 Decimal expansion of the Dirichlet eta function at 6.
%C A275703 It appears that each sum of a Dirichlet eta function is 1/2^(x-1) less than the zeta(x), where x is a positive integer > 1. In this case, eta(x) = eta(6) = (31/32)*zeta(6) = 31*(Pi^6)/30240.  Therefore eta(6) = 1/2^(6-1) or 1/32nd less than zeta(6) (see A013664). [Edited by _Petros Hadjicostas_, May 07 2020]
%F A275703 eta(6) = 31*(Pi^6)/30240 = 31*A092732/30240 = Sum_{n>=1} (-1)^(n+1)/n^6.
%F A275703 eta(6) = lim_{n -> infinity} A136677(n)/A334605(n). - _Petros Hadjicostas_, May 07 2020
%e A275703 31*(Pi^6)/30240 = 0.9855510912974...
%t A275703 RealDigits[31*(Pi^6)/30240,10,100]
%o A275703 (Sage) s = RLF(0); s
%o A275703 RealField(110)(s)
%o A275703 for i in range(1, 10000): s -= (-1)^i / i^6
%o A275703 print(s) # _Terry D. Grant_, Aug 05 2016
%Y A275703 Cf. A002162 (decimal expansion of value at 1), A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A267316 (value at 5), A275710 (value at 7).
%Y A275703 Cf. A013664, A136677, A334605.
%K A275703 nonn,cons
%O A275703 0,1
%A A275703 _Terry D. Grant_, Aug 05 2016