Clay Mathematics Institute 2005 Summer School: On Ricci Flow and the Geometrization of 3–manifolds

Summer School 2005


Clay Mathematics Institute 2005 Summer School. June 20 – July 15 at the Mathematical Sciences Research Institute (MSRI) Berkeley, California

Overview

The Clay Mathematics Institute will hold its 2005 summer school at the Mathematical Sciences Research Institute (MSRI) in Berkeley, California.
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A Problem about Limits of Sequence

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 $x_1=C/2$, where $C$ is a constant, try to show that

  1. If $C>1$, then $\set{x_n}_{n=1}^\infty$ is divergent;
  2. If $0< C\leq 1$, then $\set{x_n}_{n=1}^\infty$ is convergent;
  3. If $-3\leq C < 0$, then $\set{x_n}_{n=1}^\infty$ is convergent;

Try to discuss the case of $C< -3$, is $\set{x_n}_{n=1}^\infty$ divergent?

If you have any idea, please tell me! Just leave a word below!