Question

How do you solve for `x in RR` the equation `x! = e^x` ?

Answer 1

I got `x~=5.290316`

Explanation:

I used the extension `x! = Gamma (x+1)` iteratively in the range suggested by Cesareo R. to come up with 5.290316

Obviously, equation is also satisfied for `x=0` by definition.

Answer 2

Agree `x=5.290316...` is a solution using the Gamma function, but by the nature of the question I assumed we were looking for integer solutions.

As Gamma grows faster then the exponential I would assume this is the only solution.

graph{(y-x!)(y-e^x)=0 [-10, 10, -50, 300]}

I'll post an asymptotic solution if I get time.