$\newcommand{\set}[1]{\left\{#1\right\}}$ I have been asked to solve the following problem: Let $\set{x_n}_{n=1}^\infty$ be a real sequence defined by $$x_{n+1}=\frac{C}{2}+\frac{x_n^2}{2},$$ with $x_1=\frac{C}{2}$, where $C$ is a constant. Try to show that
- If $C>1$, then $\set{x_n}_{n=1}^\infty$ is divergent;
- If $0< C\leq 1$, then $\set{x_n}_{n=1}^\infty$ is convergent;
- If $-3\leq C < 0$, then $\set{x_n}_{n=1}^\infty$ is convergent;
For the case of $C< -3$, when is $\set{x_n}_{n=1}^\infty$ divergent?
the original link were here: A Problem about Limits of Sequence