A132326 Decimal expansion of Product_{k>=1} (1+1/10^k).
1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 1, 3, 7, 0, 5, 0, 6, 3, 2, 1, 2, 6, 0, 7, 8, 0, 6, 7, 0, 9, 4, 4, 0, 5, 8, 0, 3, 7, 4, 7, 5, 0, 7, 4, 6, 7, 5, 7, 7, 5, 9, 2, 8, 3, 5, 7, 8, 7, 9, 5, 8, 2, 3, 7, 0, 3, 3, 2, 5, 3, 4, 6, 9, 4, 8, 8, 1, 4, 1, 1, 0, 4, 3, 7, 6, 4, 7, 2, 2, 2, 2, 6, 4, 2, 1, 3, 5, 2, 3, 5, 5, 6, 4, 7, 4
Offset: 1
Examples
1.1122345691370506321260780670944...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1200
- Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.
Crossrefs
Programs
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Mathematica
digits = 105; NProduct[1+1/3^k, {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *) N[QPochhammer[-1/10,1/10]] (* G. C. Greubel, Dec 01 2015 *) -
PARI
prodinf(k=1, 1+1/10^k) \\ Amiram Eldar, May 20 2023
Formula
Equals (1/2)*lim sup_{n->oo} Product_{0<=k<=floor(log_10(n))} (1+1/floor(n/10^k)).
Equals (1/2)*lim sup_{n->oo} A132271(n)/n^((1+log_10(n))/2).
Equals (1/2)*lim sup_{n->oo} A132272(n)/n^((log_10(n)-1)/2).
Equals exp(Sum_{n>0} 10^(-n)*Sum_{k|n} -(-1)^k/k) = exp(Sum_{n>0} A000593(n)/(n*10^n)).
Equals (-1/10; 1/10){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 01 2015
Equals (sqrt(2)/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
Comments