OFFSET
0,1
COMMENTS
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]
FORMULA
eta(6) = 31*(Pi^6)/30240 = 31*A092732/30240 = Sum_{n>=1} (-1)^(n+1)/n^6.
EXAMPLE
31*(Pi^6)/30240 = 0.9855510912974...
MATHEMATICA
RealDigits[31*(Pi^6)/30240, 10, 100]
PROG
(Sage) s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s -= (-1)^i / i^6
print(s) # Terry D. Grant, Aug 05 2016
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Terry D. Grant, Aug 05 2016
STATUS
approved