Maxima of $f(x)/e^x$ where $f(x)$ is an approximation of $e^x$ using Stirling's

Question

Let $$f(x)=1+\sum_{n=1}^\infty\frac{x^n}{\sqrt{2\pi n}(n/e)^n}\tag1$$ and let $$g(x)=\frac{f(x)}{e^x}\tag2$$ If we plot $g(x)$ we get a graph that looks like this:

Clearly there is a maximum at around $x_0=1.5$, investigating further we find an approximation to this $x_0$ is $$x_0\approx 1.501418665538103821742229435035476066021$$ I did not manage to find a closed form using WolframAlpha or ISC.

Questions:

1. Has a crude approximation to $e^x$ like this ever been studied? Specifically one using stirlings approximation.
2. Although I doubt it, is there any observation that can be mind that might hint at the actual value of $x_0$?
3. Is there a good way of numerically approximating $x_0$?