A132325 - OEIS

%I #18 Aug 06 2024 05:10:53

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

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

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

%N Decimal expansion of Product_{k>=0} (1+1/10^k).

%C Twice the constant A132326.

%H G. C. Greubel, <a href="/A132325/b132325.txt">Table of n, a(n) for n = 1..1200</a>

%H Richard J. McIntosh, <a href="https://doi.org/10.1112/jlms/51.1.120">Some Asymptotic Formulae for q-Hypergeometric Series</a>, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; <a href="https://citeseerx.ist.psu.edu/pdf/4f03a5e304ec19f8a725774525aecd2a78f4ad81">alternative link</a>.

%F Equals lim sup_{n->oo} Product_{0<=k<=floor(log_10(n))} (1+1/floor(n/10^k)).

%F Equals lim sup_{n->oo} A132271(n)/n^((1+log_10(n))/2).

%F Equals lim sup_{n->oo} A132272(n)/n^((log_10(n)-1)/2).

%F Equals 2*exp(Sum_{n>0} 10^(-n)*Sum_{k|n} -(-1)^k/k) = 2*exp(Sum_{n>0} A000593(n)/(n*10^n)).

%F Equals lim sup_{n->oo} A132271(n+1)/A132271(n).

%F Equals 2*(-1/10; 1/10)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Dec 02 2015

%F Equals sqrt(2) * exp(log(10)/24 + Pi^2/(12*log(10))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(10))) (McIntosh, 1995). - _Amiram Eldar_, May 20 2023

%e 2.22446913827410126425215613418881160749501...

%t digits = 105; NProduct[1+1/10^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 18 2014 *)

%t 2*N[QPochhammer[-1/10,1/10]] (* _G. C. Greubel_, Dec 02 2015 *)

%o (PARI) prodinf(x=0, 1+(1/10)^x) \\ _Altug Alkan_, Dec 03 2015

%Y Cf. A081845, A067080, A100220, A132019-A132026, A132034-A132038, A132323-A132326, A132271, A132272, A000593.

%K nonn,cons

%O 1,1

%A _Hieronymus Fischer_, Aug 20 2007