OFFSET
1,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
Árpád Baricz, Mills' ratio: Monotonicity patterns and functional inequalities, Journal of Mathematical Analysis and Applications, vol.340, no.2, Apr 15 2008, pp.1362-1370
Iosif Pinelis, Monotonicity properties of the relative error of a Padé approximation for Mills’ ratio, Journal of Inequalities in Pure and Applied Mathematics 01/2002; 3.
Eric Weisstein's MathWorld, Mills ratio
FORMULA
The unique positive root of the transcendent equation x*(x^2+2)*r(x) = x^2+1, where r(x) is Mills' ratio exp(x^2/2)*sqrt(Pi/2)*erfc(x/sqrt(2)).
r(0) = sqrt(Pi/2).
Asymptotic expansion: r(x) ~ 1/x - 1/x^3 + 3/x^5 - 3*5/x^7 + ... + (-1)^k*(2k-1)!!/x^(2k+1) + ...
EXAMPLE
1.16152788927447736700459503109631790552481734300963234247699554...
MATHEMATICA
r[x_] := Exp[x^2/2]*Sqrt[Pi/2]*Erfc[x/Sqrt[2]]; x0 = x /. FindRoot[x*(x^2+2)*r[x] == (x^2+1), {x, 1}, WorkingPrecision -> 102]; RealDigits[x0] // First
PROG
(PARI) solve(x=1, 2, x*(x^2+2)*exp(x^2/2)*sqrt(Pi/2)*erfc(x/sqrt(2))-x^2-1) \\ Charles R Greathouse IV, Apr 30 2015
(Python)
from mpmath import *
mp.dps=103
print([int(n) for n in list(str(findroot(lambda x: x*(x**2+2)*exp(x**2/2)*sqrt(pi/2)*erfc(x/sqrt(2))-x**2-1, (1, 2))).replace('.', ''))]) # Indranil Ghosh, Jul 07 2017
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Apr 28 2015
STATUS
approved