1. ## sequence problems

#1
Prove the sequence $\displaystyle \{x_n\}$ defined in the following is increasing and bounded.
$\displaystyle x_n=1+\frac{x_{n-1}}{1+x_{n-1}}$

I suppose to use MI to prove and let S(n) by the statement
$\displaystyle x_{n+1}\geq x_{n}$ and $\displaystyle 1\leq x_n<2$
but i cant prove the sequence is increasing for the case n=k+1, am i starting with wrong statement ?

#2
Set $\displaystyle a_1=1$ ,and for $\displaystyle n\geq2$, $\displaystyle a_{n+1}=\frac{1}{2+a_n}$.Show that the sequence $\displaystyle \{a_n\}$ is convergent.

I am confusing how to let the statement.

2. ## Re: sequence problems

Originally Posted by maoro
#1
Prove the sequence $\displaystyle \{x_n\}$ defined in the following is increasing and bounded...

$\displaystyle x_n=1+\frac{x_{n-1}}{1+x_{n-1}}$
The 'attack strategy' for this type of problems is illustrated in...

http://www.mathhelpforum.com/math-he...-i-188482.html

The recurrence relation defining the $\displaystyle x_{n}$ can be written as...

$\displaystyle \Delta_{n}= x_{n+1}-x_{n} = \frac{1+x_{n} - x_{n}^{2}} {1+x_{n}} = f(x_{n})$ (1)

The function f(x) has one 'attractive fixed point' in $\displaystyle x_{+}= \frac{1+\sqrt{5}}{2}$ and one 'repulsive fixed point' in $\displaystyle x_{-}= \frac{1-\sqrt{5}}{2}$ and because the inequality $\displaystyle |f(x)|< |x_{+}-x|$ all the initial values $\displaystyle x_{0}> x_{-}$ will produce a sequence converging to $\displaystyle x_{+}$. All the initial values $\displaystyle x_{0}< x_{-}$ however will produce a diverging sequence...

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$

P.S. An important detail: all the initial values $\displaystyle x_{-}<x_{0}<x_{+}$ will produce an increasing sequence and all the initial values $\displaystyle x_{0}>x_{+}$ will produce a decreasing sequence...

3. ## Re: sequence problems

Originally Posted by maoro
#2
Set $\displaystyle a_1=1$ ,and for $\displaystyle n\geq2$, $\displaystyle a_{n+1}=\frac{1}{2+a_n}$.Show that the sequence $\displaystyle \{a_n\}$ is convergent.

I am confusing how to let the statement.
The 'strategy' is the same illustrated in my previous post. The difference equation generating the $\displaystyle a_{n}$ can be written as...

$\displaystyle \Delta_{n}= a_{n+1}-a_{n}= \frac{1}{2+a_{n}}-a_{n}=f(a_{n})$ (1)

Here f(x) has two 'attractive fixed points' in $\displaystyle x_{-}=-1-\sqrt{2}$ and $\displaystyle x_{+}=-1+\sqrt{2}$. The attractive point $\displaystyle x_{-}$ however is 'pratically unarrivable' so that we indagate on $\displaystyle x_{+}$. For this 'attractive fixed point' the conditions of convergence are satisfied for any 'initial value' $\displaystyle a_{0}>-2$. For $\displaystyle -2<a_{0}<x_{+}$ the convergence is 'oscillatory' and for $\displaystyle a_{0}>x_{+}$ the convergence will be 'monotonic'...

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$

4. ## Re: sequence problems

sorry, but are there easier alternatives to approach them ?