A267315 Decimal expansion of the Dirichlet eta function at 4.

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, 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, 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

Offset: 0

Author

Ilya Gutkovskiy, Jan 13 2016

Keywords

• nonn
• cons

Examples

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

Links

• OEIS Wiki, Euler's alternating zeta function.
• Eric Weisstein's World of Mathematics, Dirichlet Eta Function.
• Wikipedia, Dirichlet Eta Function.

Crossrefs

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

Programs

Magma

pi:= 7*Pi(RealField(110))^4 / 720; Reverse(Intseq(Floor(10^100*pi))); // Vincenzo Librandi, Feb 04 2016

Mathematica

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

PARI

7*Pi^4/720 \\ Michel Marcus, Feb 01 2016

Sage

s = RLF(0); s
RealField(110)(s)
for i in range(1,10000): s += -((-1)^i/((i)^4))
print(s) # Terry D. Grant, Aug 04 2016

Formula

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

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