%I #41 Jan 30 2022 15:57:20

%S 6,8,2,1,5,3,5,0,2,6,0,5,2,3,8,0,6,6,7,6,1,2,6,3,1,8,6,2,2,6,6,2,4,0,

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

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

%N Decimal expansion of Sum_{n>=1} 1/(3^n-1).

%H G. C. Greubel, <a href="/A214369/b214369.txt">Table of n, a(n) for n = 0..10000</a>

%F Equals Sum_{n>=1} 1/A024023(n).

%F Equals Sum_{k>=1} d(k)/3^k, where d(k) is the number of divisors of k (A000005). - _Amiram Eldar_, May 17 2020

%e Equals 0.6821535026052380667...

%p evalf(sum(1/(3^k-1), k=1..infinity), 120); # _Vaclav Kotesovec_, Oct 18 2014

%p # second program with faster converging series

%p evalf( add( (1/3)^(n^2)*(1 + 2/(3^n - 1)), n = 1..14 ), 105); # _Peter Bala_, Jan 30 2022

%t RealDigits[ NSum[1/(3^n - 1), {n, 1, Infinity}, WorkingPrecision -> 110, NSumTerms -> 100], 10, 105] // First (* or *) 1 - (Log[2] + QPolyGamma[0, 1, 1/3])/Log[3] // RealDigits[#, 10, 105]& // First (* _Jean-François Alcover_, Jun 05 2013 *)

%t x = 1/3; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* _Robert G. Wilson v_, Oct 12 2014 after an observation and the formula of _Amarnath Murthy_, see A073668 *)

%o (PARI) suminf(n=1, 1/(3^n-1)) \\ _Michel Marcus_, Mar 11 2017

%Y Cf. A024023, A065442, A073668, A248721, A248722, A248723, A248724, A248725, A248726.

%K cons,nonn

%O 0,1

%A _R. J. Mathar_, Jul 14 2012

%E More terms from _Jean-François Alcover_, Feb 12 2013