The usual approximation is
which is just the beginning of an asymptotic series
and we can rewrite the square root to get
and combining the two square roots gives us
which is what we wanted. This is basically a combination of the first two terms of the usual Stirling approximation into one term. For example, when n = 8 this gives 40316, compared to 39902 for the first term of Stirling's series and 40318 for the first two. (8! = 40320.) In general the error of Gosper's approximation is about twice that in the first two terms of Stirling's, so this isn't incredibly useful but it's an interesting curiosity.
