An Asymptotic Form of the Generating Function ${\prod }_{k=1}^{\infty }(1+x^k/k)$

Abstract

It is shown that the sequence of rational numbers $r(k)$ generated by the ordinary generating function ${\prod }_{k=1}^{\infty }(1+x^k/k)$ converges to a limit $C>0$. $C$ can be expressed as $C=\exp (-{\sum }_{k=2}^{\infty }\frac{(-1{)}^k}{k}\zeta (k))$ where $\zeta ()$ denotes the Riemann zeta function.

9 pages, 1 figure