%I #26 Jan 30 2022 15:58:04
%S 1,3,9,0,4,5,1,1,7,6,6,2,1,8,8,1,2,9,3,5,8,7,2,8,4,7,4,3,6,9,0,8,9,0,
%T 5,2,1,3,9,3,6,2,6,4,7,0,6,7,8,1,9,6,0,9,5,5,1,0,3,5,4,9,3,4,7,9,6,7,
%U 0,2,0,1,4,5,3,6,6,6,6,0,7,9,8,8,6,3,3,7,9,8,1,3,5,7,6,5,5,0,5,7,9,9,5,5,3
%N Decimal expansion of Sum_{k>=1} 1/(9^k - 1).
%H G. C. Greubel, <a href="/A248726/b248726.txt">Table of n, a(n) for n = 0..10000</a>
%F Equals Sum_{k>=1} d(k)/9^k, where d(k) is the number of divisors of k (A000005). - _Amiram Eldar_, Jun 22 2020
%e 0.13904511766218812935872847436908905213936264706781960955103549347967020145366...
%p evalf(sum(1/(9^k-1), k=1..infinity),120) # _Vaclav Kotesovec_, Oct 18 2014
%p # second program with faster converging series
%p evalf( add( (1/9)^(n^2)*(1 + 2/(9^n - 1)), n = 1..10), 105); # _Peter Bala_, Jan 30 2022
%t x = 1/9; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* after an observation and the formula of _Amarnath Murthy_, see A073668 *)
%o (PARI) suminf(k=1, 1/(9^k-1)) \\ _Michel Marcus_, Oct 18 2014
%Y Cf. A000005, A065442, A073668, A214369, A248721, A248722, A248723, A248724, A248725.
%K nonn,cons
%O 0,2
%A _Robert G. Wilson v_, Oct 12 2014