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A275703
Decimal expansion of the Dirichlet eta function at 6.
7
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, 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, 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
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.
eta(6) = lim_{n -> infinity} A136677(n)/A334605(n). - Petros Hadjicostas, May 07 2020
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
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).
Sequence in context: A371219 A256165 A345737 * A094141 A377602 A200292
KEYWORD
nonn,cons
AUTHOR
Terry D. Grant, Aug 05 2016
STATUS
approved