Problem :
Let :
$$\int_{0}^{\infty}\left(\frac{\left(1+\sqrt{5}\right)}{2}\right)^{-x!}x^{2}dx<\frac{\pi^{2}}{6}\frac{\left(1+\sqrt{5}\right)}{2}$$
It's a crude estimation so I think it's doable by hand .
I have tried an intregration by part but nothing comes .
I can show the convergence as :
for $x>0$ we have :
$$x!>x/2$$
Proof :
On $(0,1)$ we have $x!>x>x/2$ and on $(1,\infty)$ we have $f(x)=x!>f'(1)(x-1)+f(1)>x/2$ because the Gamma function is convex .
Wich gives a result around $143.59$
My progress so far :
Using the lemma 7.1 (see reference) we have for $0<x<\frac{\left(1+\sqrt{5}\right)}{2}$:
$$\left(\left(1-x!\right)^{2}+\left(\frac{\left(1+\sqrt{5}\right)}{2}\right)^{-1}x!(2-x!)-\left(\frac{\left(1+\sqrt{5}\right)}{2}\right)^{-1}x!(1-x!)\ln\left(\left(\frac{\left(1+\sqrt{5}\right)}{2}\right)^{-1}\right)\right)x^{2}\geq \left(\frac{\left(1+\sqrt{5}\right)}{2}\right)^{-x!}x^2=f(x)$$
And numerically it seems we have for $12/5<x$ :
$$\frac{f\left(2\right)}{g\left(2\right)}g\left(x+\frac{39}{1000}\right)>\left(\frac{\left(1+\sqrt{5}\right)}{2}\right)^{-x!}x^{2}$$
Where :
$$g\left(x\right)=e^{1-\left(x-2\right)^{e}}$$
How to show it by hand without a computer ?
Reference :
VASILE CIRTOAJE, PROOFS OF THREE OPEN INEQUALITIES WITH POWER-EXPONENTIAL FUNCTIONS, Journal of Nonlinear Sciences and Applications, 4 (2011), no. 2, 130-137