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 ?