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

0,1

References

Konrad Knopp, Theory and application of infinite series, Blackie & Son Limited, London and Glasgow, 1954. See p. 239.

Links

Table of n, a(n) for n=0..119.

OEIS Wiki, Euler's alternating zeta function.

Eric Weisstein's World of Mathematics, Dirichlet Eta Function.

Wikipedia, Dirichlet Eta Function.

Formula

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

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

Example

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

Mathematica

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

Prog

(PARI) 7*Pi^4/720 \\ Michel Marcus, Feb 01 2016
(Magma) pi:= 7*Pi(RealField(110))^4 / 720; Reverse(Intseq(Floor(10^100*pi))); // Vincenzo Librandi, Feb 04 2016
(SageMath) 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

Crossrefs

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

Sequence in context: A381696 A131109 A371881 * A247412 A154397 A116186

Adjacent sequences: A267312 A267313 A267314 * A267316 A267317 A267318

Keyword

nonn,cons

Author

Ilya Gutkovskiy, Jan 13 2016

Status

approved