# Simplifying the inequality $nx(1-x)^n \leq \epsilon$ to $n > N(x, \epsilon)$?

How can I reduce the following inequality on $$x \in [0, 1]$$, and $$\epsilon > 0$$: $$nx(1-x)^n \leq \epsilon$$ to the this form $$n > N(x, \epsilon)\quad?$$ We can assume $$n = 1, 2, 3, \dots$$.

My attempt:

$$\ln\big(nx(1-x)^n\big) \leq \ln \epsilon$$

$$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \implies \ln n + n\ln(1-x) \leq \ln\frac{\epsilon}{x}$$,

and I am stuck. Since $$\ln(1-x) < 0$$, I think I must divide by it to change the direction of inequality. But I don't know how to simplify/group $$n$$ and $$\ln n$$ while decoupling them from $$x$$ term.

Ultimately, I want to prove that $$\forall n \geq N(x, \epsilon), \quad nx(1-x)^n \leq \epsilon$$. Perhaps, I do not need to solve for $$n$$?

• Are you really trying to manipulate the inequality to obtain $n>N(x,\varepsilon)$? Or maybe you just need to find $N(x,\varepsilon)$ such that for $n\ge N(x,\varepsilon)$ your original inequality holds? Commented Jan 29, 2021 at 10:29
• @TitoEliatron, sorry, I edited minutes ago. Yes, that's what I want ultimately. But just finding $N(x, \epsilon)$ doesn't seem straightforward to me either, so I wanted to solve for it.
– js9
Commented Jan 29, 2021 at 10:32

As a real, the solution of $$nx(1-x)^n = \epsilon$$ is given in terms of Lambert function $$n=\frac{1}{\log (1-x)}W\left(\frac{ \log (1-x)}{x}\epsilon \right)$$

Viewing your last sentence, you can procceed as follows.

Take $$\varepsilon>0$$ (WLOG, you can assume that $$\varepsilon<1$$).

for $$x=0,1$$, it is clear that $$nx(1-x)^n=0<\varepsilon$$ for all $$n$$.

For $$0, since $$n(1-x)^n\to 0$$ ($$n\to\infty$$) (recall that $$0<1-x<1$$) you can find $$N=N(x,\varepsilon)$$ such that if $$n\ge N$$, $$n(1-x)^n<\varepsilon$$.

Now $$nx(1-x)^n\le n(1-x)^n<\varepsilon$$, since $$x\le 1$$.

• It seems that he is trying to prove that $nx(1-x)^n \to 0$ as $n \to +\infty$, so assuming that $n(1-x)^n \to 0$ seems a bit too much (not that it is a hard fact to prove; it simply does arise from immediate properties) Commented Jan 29, 2021 at 10:41