1. ## analysis

Consider the sequence $x_n \mbox {defined by x_1 = \alpha(\alpha \in R fixed) , x_{n+1} =\frac{1}{4}(1+x_n), n \geq 1 . }$
$\mbox { Determine when x_n is increasing or decreasing , then prove the convergence of x_n$

2. Originally Posted by flower3
Consider the sequence $x_n \mbox {defined by x_1 = \alpha(\alpha \in R fixed) , x_{n+1} =\frac{1}{4}(1+x_n), n \geq 1 . }$
$\mbox { Determine when x_n is increasing or decreasing , then prove the convergence of x_n$
$x_{n+1}=\frac{1}{4}(1+x_n)\Rightarrow x_{n+1}-\frac{1}{3}=\frac{1}{4}(x_n-\frac{1}{3})$
So $x_n-\frac{1}{3}=\frac{1}{4^{n-1}}(x_1-\frac{1}{3})=\frac{1}{4^{n-1}}(\alpha-\frac{1}{3})$
So $x_n=\frac{1}{3}+\frac{1}{4^{n-1}}(\alpha-\frac{1}{3})$
so $\{x_n\}$converges for every $\alpha$