Rational approximants for the Euler-Gompertz constant
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 δ is <ln(x+1)>. Main idea. Let Pn(x) be a polynomial with integer coefficients. It is easy to prove that <Pn(x)ln(x+1)>=an+<ln(x+1)>bn$ for some integers an, bn. Hence if <Pn(x)ln(x+1)>/bn converges to zero, an/bn converges to - δ . Main Theorem. Let u be positive real. There exists polynomials Pn(x)(they are explicitly given in the paper) such that <Pn(x)ln(xu+1)> tends to u as n tends to infinity. Proof of Main Theorem is elementary.
Turn this paper into a lesson
ArcXiv compiles a structured reading guide from this paper's metadata: plain-English importance, contributions, prerequisite concepts, which sections to read first, flashcards, and a quiz. Grounded in the abstract, never invented.