The wording of this question is a bit clunky, so allow me to clarify it. What I mean by integer factorials is n!, where n is an integer, more specifically a natural number. So this includes 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, and so on.
The reciprocals would be 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/362880, and so on. 1 more than each of these reciprocals would give us 2, 3/2, 7/6, 25/24, 121/120, 721/720, 5041/5040, and so on. Then you'd multiply each of these numbers together to get the infinite product we want. That is,
(2)(3/2)(7/6)(25/24)(121/120)*(721/720)...
This value seems to be roughly 3.6182154. I'd like to know it's exact value, like if it is some combination of square roots, Euler's constant, and/or pi, and why it is that value.
