A085361 Decimal expansion of the number c = Sum_{n>=1} (zeta(n+1)-1)/n.

7, 8, 8, 5, 3, 0, 5, 6, 5, 9, 1, 1, 5, 0, 8, 9, 6, 1, 0, 6, 0, 2, 7, 6, 3, 2, 3, 4, 5, 4, 5, 5, 4, 6, 6, 6, 4, 7, 2, 7, 4, 9, 6, 6, 8, 2, 2, 3, 2, 8, 1, 6, 4, 9, 7, 5, 5, 1, 5, 6, 4, 0, 2, 3, 0, 1, 7, 8, 0, 6, 4, 3, 5, 6, 3, 3, 0, 1, 6, 2, 2, 8, 7, 4, 7, 1, 5, 9, 2, 1, 3, 3, 2, 2, 4, 3, 1, 9, 6, 7, 5, 6

Offset

0,1

Comments

The Alladi-Grinstead constant (A085291) is exp(c-1).

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 528 and 538.

Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Vol. 29, No. 2 (June 1988), pp. 196-205.

Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant.

Eric Weisstein's World of Mathematics, Convergence Improvement.

Formula

Equals Sum_{n>=2} log(n/(n-1))/n = Sum_{n>=1, k>=2} 1/(n*k^(n+1)). [From Mathworld links]

Equals -Sum_{k>=2} (-1)^k * zeta'(k). - Vaclav Kotesovec, Jun 17 2021

Equals log(A245254) = Sum_{k>=1} log(k)/(k*(k+1)). - Amiram Eldar, Jun 27 2021

Equals -log(A242624). - Amiram Eldar, Feb 06 2022

Example

0.78853056591150896106027632345455466647274966822328164975515640230178...

Maple

evalf(sum((Zeta(n+1)-1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015
evalf(Sum(-(-1)^k*Zeta(1, k), k = 2..infinity), 120); # Vaclav Kotesovec, Jun 18 2021

Mathematica

Sum[(-1+Zeta[1+n])/n, {n, Infinity}]
NSum[Log[k]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms ->5000, Method -> {NIntegrate, MaxRecursion -> 100}] (* Vaclav Kotesovec, Dec 11 2015 *)

Prog

(PARI) suminf(n=1, (zeta(n+1)-1-2^(-n-1))/n)+log(2)/2 \\ Charles R Greathouse IV, Feb 20 2012
(PARI) sumalt(k=2, -(-1)^k * zeta'(k)) \\ Vaclav Kotesovec, Jun 17 2021
(Sage)
import mpmath
mpmath.mp.pretty=True; mpmath.mp.dps=108 #precision
mpmath.nsum(lambda n: (-1+mpmath.zeta(1+n))/n, [1, mpmath.inf]) # Peter Luschny, Jul 14 2012
(Sage) numerical_approx(sum((zeta(k+1)-1)/k for k in [1..1000]), digits=120) # G. C. Greubel, Nov 15 2018
(Magma) SetDefaultRealField(RealField(120)); L:=RiemannZeta(); (&+[(Evaluate(L, n+1)-1)/n: n in [1..1000]]); // G. C. Greubel, Nov 15 2018

Crossrefs

Cf. A002210, A085291, A242624, A244109, A245254.

Sequence in context: A353781 A338815 A197810 * A256781 A248224 A092290

Adjacent sequences: A085358 A085359 A085360 * A085362 A085363 A085364

Keyword

nonn,cons

Author

Eric W. Weisstein, Jun 25 2003

Status

approved