Wikipedia:Reference desk/Archives/Mathematics/2013 July 6

= July 6 =

Convergence and Closed Form Expression

 * $$\int_0^\infty{b(x) \over B(x)}\ dx\ =\ ?\ ,\qquad\qquad b(x) = \sum_{n = 1}^\infty {n^x \over n^n}\ ,\qquad\qquad B(x) = \sum_{n = 1}^\infty {n^x \over n!}$$

Observations:
 * 1. B(x) is e times a Bell number.
 * 2. If the numerator of each function would be xn instead of nx, and each sum would start at 0 instead of 1, then the value of our integral would become $$\sum_{n = 0}^\infty{n! \over n^n}\ ,$$ where lim nn → 1 when n → 0.
 * — 79.113.210.135 (talk) 20:45, 6 July 2013 (UTC)

You could try to use Ramanujan's master theorem. Count Iblis (talk) 21:40, 6 July 2013 (UTC)
 * In that case, s = 1 and $$f(x) = \frac{b(x)}{B(x)}\ ,$$ whose Taylor expansion is absolutely catastrophic... 79.113.210.135 (talk) 23:11, 6 July 2013 (UTC)
 * To make matters even more complicated than they already are, integration by parts would seem to suggest that the integral is divergent if any of the two sums starts from n = 1, despite numerical data indicating the contrary... The question now would be whether starting at n = 2 would really make that much of a difference anyway... 79.113.225.68 (talk) 13:58, 7 July 2013 (UTC)