Convergence of the recursive sequence $x_n = x_{n-1} - \frac12\lambda(x_{n-1}-1)^{-3/2}$

Question

Let $\lambda \geq 0$ and define the recursive sequence $$ x_n = x_{n-1} - \frac{\lambda}{2(x_{n-1}-1)^{\frac{3}{2}}} $$ I know that this sequence is bounded below by 1 and I also need to determine a starting point $x_0$ such that the serie converges (maybe expressed as a function of $\lambda$). I start setting up the problem, by using the inequality $$ 1 \leq x - \frac{\lambda}{2(x-1)^{\frac{3}{2}}} \leq x $$ However, I have absolutely no idea about how to handle such inequality. Thanks to anybody willing to help! :)

Answer 1

Hint

as we have

$$x_{n}=x_{n-1}-\frac{\lambda}{2(x_{n-1}-1)^\frac32}$$

if $(x_n)$ converges to $L$ then

we will have

$$L=L-\frac{\lambda}{2(L-1)^\frac32}$$

impossible if $\lambda>0$ and $(x_n)$ diverges.

if $\; \lambda=0\; \;x_n=x_0=C^t$