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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 n^x, 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)[reply]

You could try to use Ramanujan's master theorem. Count Iblis (talk) 21:40, 6 July 2013 (UTC)[reply]

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)[reply]

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)[reply]

Unresolved