In a formula in my self-study of a summation method based on the matrix of Eulerian numbers (which I thus call "Eulerian summation") I am considering terms like $$ \Big({n \over e}\Big)^n \cdot {1 \over n!} \cdot {1 \over n+1} \tag 1 $$ For the estimate of the growth there is the Stirling-approximation-formula for the factorial: $$ m! \approx \sqrt{2 \pi m} \Big({m \over e}\Big)^m \tag 2$$
so that I arrive at the estimate
$$ \Big({n \over e}\Big)^n {1 \over n!} \approx {n! \over \sqrt{2 \pi n} }\cdot {1 \over n!} = {1 \over \sqrt{2 \pi n} } \tag 3 $$
(where I omit here and in the following the $ {1 \over n+1}$-term)
But by inspection of some results with increasing n I found, that a much better approximation than(2) is
$$ \Big({n \over e}\Big)^n \approx \Gamma(1+n-1/2) \cdot {1 \over \sqrt{2 \pi }} \tag 4$$
From this, and from the coincidence that $\sqrt{\pi} = \Gamma(1/2)=\Gamma(1-1/2)$ one might then write
$$ \Big({n \over e}\Big)^n {1 \over n!}
\approx {\Gamma(1+n-1/2) \over \Gamma(1-1/2) \cdot \Gamma(1+n) }\cdot {1 \over \sqrt{2}}\\ = \binom{n-1/2}{n}\cdot {1 \over \sqrt{2} } \tag 5
$$
which is then a simple (generalized) binomial-expression.
By comparision , the series
$$ s = \sum_{n=0}^\infty \Big({n \over e}\Big)^n {1 \over n!} \cdot {1 \over n+1} \tag 6$$
should then be a convergent expression and nicely approximated by.
$$ t = {1\over \sqrt 2}\sum_{n=0}^\infty \binom{n-1/2}{n} \cdot {1 \over n+1} \sim s\tag 7$$
I thought, after that improvement the formula (4) might be even more improvable, so I looked first at Peter Luschny's "factorial" site where I find a related expression at the paragraph "A simple expression(recommended)" with the "LuschnyCF4" where however $m-1/2$ is replaced by $n$ and $n+1/2$ by $N$ and the approximation reads then as $$ n! \approx \sqrt{2 \pi } \Big({P \over e}\Big)^N \\ \text{where } N=n+1/2 \tag 8$$ Here $P$ is a complicated scaling of $N$ so I cannot simply adapt that formula for an improved approximation. So my question is:
Q: How can I improve my (heuristical) approximation in the rhs in (4) (while keeping it focused at positive integers $n$ in the term $(n/e)^n$)?
(P.s.: I did not have a good idea for tagging, perhaps someone could improve that)