How can it be proved that
$$ f(x) = \frac{\sum_{n=0}^{\infty} \frac{n}{2 x} \frac{(\sqrt{x})^{n}}{\sqrt{n!}}}{\sum_{n=0}^{\infty} \frac{(\sqrt{x})^{n}}{\sqrt{n!}}} $$
is monotonic?
The plot of $f(x)$ shows that it is strictly decreasing, and this is due to the $x$ in the denominator of the upper summation. (Was it not for that $x$, $f(x)$ would be increasing and there is a theorem that states this.)