**1.** *Does the following integral converge ?*  
  



$$\int\_0^\infty \frac{b(x)}{B(x)} dx$$


*where*


$$b(x) = \sum\_{n=1}^\infty \frac{n^x}{n^n} \qquad and \qquad B(x) = \sum\_{n=1}^\infty \frac{n^x}{n!}$$


  
**2.** *Does it possess a closed form, or some other alternative expression ?*



Convergence and Closed Form of an Integral Involving Bell Numbers
The question on convergence is certainly not research level.
I'm too lazy to give a detailed answer, but here is a direction in which I believe one can obtain a proof with a little effort.


Obviously, all we need is to estimate the asymptotics of $B(x)$ and $b(x)$ for $x \to \infty$. Using a variant of [Laplace's method](http://en.wikipedia.org/wiki/Laplace%27s_method) one can show that for large $x$ the value of the sums $b(x)$ and $B(x)$ is asymptotic to the maximal term in these sums, up to a polynomial factor. The maximal term is somewhere around $n(x) \sim \frac{x}{\log x}$, so $b(x) / B(x) \sim {n(x)!} / {n(x)^{n(x)}} \sim e^{-n(x)}$, up to a polynomial factor again. This $e^{-n(x)}$ is approximately $e^{-x / \log x}$, so it decays exponentially at infinity, therefore the integral converges.


