1.Does the following definite integral converge ?

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

$$\int_0^\infty \frac{b(x)}{B(x)} dx \quad = \quad ?$$

where

$$b(x) = \sum_{n=1}^\infty \frac{n^x}{n^n} \qquad and \qquad B(x) = \sum_{n=1}^\infty \frac{n^x}{n!}$$
3.If the integral may prove to be divergent when the two sums start at n = 1, could anyone check the case when the two sums start at n = 2 or 3 ?

• Thank you !