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

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 $C = \exp\Bigl(-\sum_{k = 2}^\infty \frac{(-1)^k}{k}\ \zeta(k) \Bigr)$ where $\zeta()$ denotes the Riemann zeta function.