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
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jun 25 2003
STATUS
approved