Show that a sequence converges linearly

Question

Given the sequence converges to $p^*$, show that it converges linearly: $p_{n+1}=\frac{1}{2}ln(p_n+1)$, $p_0=1$, and the limit is $p^*=0$.

I want to use fixed point theorem and denote $p_{n+1}=g(p_n)$ to show that $g'(p^*)\neq0$. But I'm not sure if it is the right approach to do so.

Thanks for any help!

Answer 1

It is easy to check the conditions of Banach's fixed point theorem, for example on the interval $[0,1]$, which proves convergence of the given sequence. On the other hand, since $g'(x)\ne 0$ in $[0,1]$, $g'(p^*)\ne 0$ and the result follows. In summary, you have a good plan!