1.Does the following definite integral converge ? 2.Does it have a closed form, or some other alternative expression ? <br><br> $$\int_0^\infty \frac{b(x)}{B(x)} dx \quad = \quad ?$$ where $$b(x) = \sum_{n=1}^\infty \frac{n^x}{n^n} \qquad and \qquad B(x) = \sum_{n=1}^\infty \frac{n^x}{n!}$$ <br> 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 !