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

1,4

Comments

Half the constant A132325.

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.

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 (1/2)*lim sup_{n->oo} A132271(n+1)/A132271(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

Example

1.1122345691370506321260780670944...

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 *)

Prog

(PARI) prodinf(k=1, 1+1/10^k) \\ Amiram Eldar, May 20 2023

Crossrefs

Cf. A000593, A067080, A079555, A132019-A132026, A132034-A132038, A132265-A132268, A132271, A132272, A132325.

Sequence in context: A091583 A245438 A245439 * A351595 A027195 A008483

Adjacent sequences: A132323 A132324 A132325 * A132327 A132328 A132329

Keyword

nonn,cons

Author

Hieronymus Fischer, Aug 20 2007

Status

approved