Upper bound on iterations of a decreasing function

Question

I have the following decreasing function:

$$n_0 =n$$ $$n_{i+1} = n_i \left ( 1 - \frac{4}{\ln n_i + 2} \right)$$

Given $1 < k \ll n$ what is a good upper bound for the number of iterations needed to reach a value $n_i < k$ ?

Answer 1

Your sequence is decreasing, so as long as $\ln n_i > 2$: $$ \begin{align*} n_i &\le n_0 \\ \frac{4}{\ln n_i + 2} &\ge \frac{4}{\ln n_0 + 2} \\ 1 - \frac{4}{\ln n_i + 2} &\le 1 - \frac{4}{\ln n_0 + 2} \\ n_i &\le n_0 \left( 1 - \frac{4}{\ln n_0 + 2} \right)^i \end{align*} $$ So the (easy) solution for $i$ to: $$ \frac{k}{n_0} = \left( 1 - \frac{4}{\ln n_0 + 2} \right)^i $$ gives an upper bound. Dunno if "good"...