How to turn bound on $ne^{-n}$ into bound for n?

Question

I have a bound of the form $ne^{-n}\leq C$, where $C$ is a constant greater than 0. I was wondering if anyone had ideas on how to turn this into a bound of the form $n\leq C'$?

Answer 1

$ne^{-n}$ has a maximum at $n=1$, so $ne^{-n}\leq C$ is always true if $C \ge e^{-1}$. Now for $0 < C < e^{-1}$, it is clear that all $n \le 0$ satisfy this condition.

For positive $n$, you can indeed find a corresponding condition but this is $n \color{red}{\ge} C'$, since $ne^{-n}$ is falling with $n$. You find $C'$ by equating $C'e^{-C'}= C$.

Now solving this equation is not possible with standard algebraic functions. There is however the Lambert W-function which adresses exactly this type of equation (see here). It is given as $y = W(x)$ where $y e^y = x$.

So you can bring it into that form by writing it as ${-C'}e^{-C'}= -C$, hence $-C' = W(-C)$ or $C' = -W(-C)$.