I would like to find $\min_{n\in\mathbb{N}} (n!\cdot x^n)$ for $0 $$
0=x^{n-1}\Gamma(n+1)\left(n+x\left[\sum_{k=1}^n\frac{1}{k}-\gamma\right] \right)
$$ however this doesn't look very good. Does anybody know a better way of tackling this problem?
Minimising a function over the positive integers
0
$\begingroup$
optimization
1 Answers
0
Hint:
Set $u_n=n! x^n$. The sequence $(u_n)$ is non-increasing as long as $\;\dfrac{u_n}{u_{n-1}}=nx\le1$, i.e. $\; n\le\biggl\lfloor\dfrac1x\biggr\rfloor$. Thus the minimum is attained at $N=\biggl\lfloor\dfrac1x\biggr\rfloor$.