It seems that $$\left(\frac{{\pi}^n}{nn!}\right)_{n\ge 1}$$ is a decreasing sequence. How can I prove it?
I noticed that replacing $\pi$ by $2\pi$ makes the sequence increasing for a bit, and removing the $n$ in front of $n!$ makes the sequence increasing for a bit as well. The problem in hand seems quite non-trivial and I'm stuck.
Write $a_n$ for the $n$th term. Then $a_{n+1}/a_n=n\pi/(n+1)^2$, and we need this to be always less than $1$. This is clearly true for $n\geq 3$ since $n\pi/(n+1)^2<\pi/(n+1)\leq \pi/4$. So we just need to check $n=1,2$ for which we get $\pi/4$ and $2\pi/9$ respectively.
(For the corresponding sequence with $2\pi$ you'd need $n\pi/(n+1)^2<1/2$, which isn't true for small values.)