Rational approximants for the Euler-Gompertz constant
  We obtain two sequences of rational numbers which converge to the Euler-Gompertz constant. Denote by <f(x)> the integral of f(x)e^{-x} from 0 to infinity. Recall that the Euler-Gompertz constant \delta is <ln(x+1)>.   Main idea. Let P_n(x) be a polynomial with integer coefficients. It is easy to prove that <P_n(x)ln(x+1)>=a_n+<ln(x+1)>b_n$ for some integers a_n, b_n. Hence if <P_n(x)ln(x+1)>/b_n converges to zero, a_n/b_n converges to - \delta .   Main Theorem. Let u be positive real. There exists polynomials P_n(x)(they are explicitly given in the paper) such that <P_n(x)ln(xu+1)> tends to u as n tends to infinity.   Proof of Main Theorem is elementary. 
Vasily Bolbachan
math.NT math.AP