A Problem about Limits of Sequence

Vanabel/ 9月 24, 2012/ 数学笔记/ 2 comments

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!

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  1. 微积分的助教真悲催啊。《数学分析中的证题方法与难题选解》第8面

    1. 哈哈, 是么? 还以为这个问题没解呢?你看, 我在strackexchange上问的同样问题, 答案也不是非常好啊.

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