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Decimal expansion of Sum_{k>=1} 1/(7^k - 1).
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%I #24 Jan 30 2022 15:57:53

%S 1,9,0,9,1,0,0,6,2,4,1,0,2,6,1,5,7,8,2,0,2,1,9,9,6,4,4,4,1,7,6,9,1,1,

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

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

%N Decimal expansion of Sum_{k>=1} 1/(7^k - 1).

%H G. C. Greubel, <a href="/A248724/b248724.txt">Table of n, a(n) for n = 0..10000</a>

%F Equals Sum_{k>=1} d(k)/7^k, where d(k) is the number of divisors of k (A000005). - _Amiram Eldar_, Jun 22 2020

%e 0.1909100624102615782021996444176911687692684760082664083347711086409996755846...

%p evalf(sum(1/(7^k-1), k=1..infinity),120) # _Vaclav Kotesovec_, Oct 18 2014

%p # second program with faster converging series

%p evalf( add( (1/7)^(n^2)*(1 + 2/(7^n - 1)), n = 1..11), 105); # _Peter Bala_, Jan 30 2022

%t x = 1/7; 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/(7^k-1)) \\ _Michel Marcus_, Oct 18 2014

%Y Cf. A000005, A065442, A073668, A214369, A248721, A248722, A248723, A248725, A248726.

%K nonn,cons

%O 0,2

%A _Robert G. Wilson v_, Oct 12 2014