Rational approximants for the Euler-Gompertz constant

TL;DR

This paper constructs rational sequences converging to the Euler-Gompertz constant using polynomial approximations, providing explicit sequences and elementary proofs of convergence.

Contribution

It introduces explicit polynomial sequences that approximate the Euler-Gompertz constant through rational convergence, extending previous methods.

Findings

Constructed two rational sequences converging to the Euler-Gompertz constant.

Provided explicit polynomials with proven convergence properties.

Demonstrated elementary proof techniques for the approximation.

Abstract

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.