Question about decreasing sequence and finding its limit

[evinda]

Hey! :)
I have a question.
It is given that $a>0 , x_{1}=x>0 \text{ and } x_{n+1}=\frac{1}{2}(x_{n}+\frac{a}{x_{n}})$ and I have to show that the sequence $(x_{n})$,at least from its second term,is decreasing and bounded from below from $\sqrt{a}$.Also,I have to find the limit $\lim_{n \to \infty}x_{n}$.

To show that the sequence is decreasing,I thought that I could take $\frac{x_{n+1}}{x_{n}}$ and show that it is smaller that $1$.I found that $\frac{x_{n+1}}{x_{n}}=\frac{1}{2}(1+\frac{a}{x_{n}^{2}})=\frac{1}{2}+\frac{a}{2x_{n}^{2}}$.
But do we know that it is $\frac{1}{2}+\frac{a}{2x_{n}^{2}}<1 \Rightarrow x_{n}>\sqrt{a}$ ?

 

[Ackbach]

If the sequence is bounded below and decreasing, then the limit exists. So, assuming you've shown these two, in order to find the limit, just take the limit of both sides of the equation:
$$\lim_{n\to \infty}x_{n+1}= \lim_{n \to \infty} \frac12 \left(x_{n}+ \frac{a}{x_{n}} \right),$$
or if $\lim_{x\to \infty}x_{n}=L$, then
$$L= \frac12 (L+a/L).$$
Now just solve for $L$.

 

[evinda]

If the sequence is bounded below and decreasing, then the limit exists. So, assuming you've shown these two, in order to find the limit, just take the limit of both sides of the equation:
$$\lim_{n\to \infty}x_{n+1}= \lim_{n \to \infty} \frac12 \left(x_{n}+ \frac{a}{x_{n}} \right),$$
or if $\lim_{x\to \infty}x_{n}=L$, then
$$L= \frac12 (L+a/L).$$
Now just solve for $L$.

I understand..Could I also say that because of the fact that the sequence is bounded below and decreasing,it converges to its inf??

Also,how can I show that it is decreasing? Can I use $\frac{x_{n+1}}{x_{n}} $ or do I have to do something else?

 

[Ackbach]

I understand..Could I also say that because of the fact that the sequence is bounded below and decreasing,it converges to its inf??

Yes, any monotonically decreasing sequence that is bounded below has to converge.

Also,how can I show that it is decreasing? Can I use $\frac{x_{n+1}}{x_{n}} $ or do I have to do something else?

You could try to show that $x_{n+1}/x_{n}$ is less than $1$. That might work. What do you get when you try that?

 

[evinda]

Yes, any monotonically decreasing sequence that is bounded below has to converge.

But we know that then the sequence converges to its infimum,or am I wrong?

You could try to show that $x_{n+1}/x_{n}$ is less than $1$. That might work. What do you get when you try that?

I found this: $\frac{1}{2}+\frac{a}{2x_{n}^{2}}$ ...But..Do we know that this is less than $1$ ?

 

[Ackbach]

But we know that then the sequence converges to its infimum,or am I wrong?

Well, it depends on $x_{1}$. You probably could say that the sequence converges to its $ \lim \inf$.

I found this: $\frac{1}{2}+\frac{a}{2x_{n}^{2}}$ ...But..Do we know that this is less than $1$ ?

I think you'll have to take three cases: $x_{1}< \sqrt{a}$, $x_{1}= \sqrt{a}$, and $x_{1}> \sqrt{a}$. The middle case is fairly straight-forward: the sequence converges to $\sqrt{a}$ immediately, and never changes. Also, the case $x_{1}> \sqrt{a}$ is straight-forward: the sequence is monotonically decreasing. It's that pesky $x_{1}< \sqrt{a}$ case that's troubling... What do you make of it?

 

[Krizalid1]

Spoiler

By induction. Let $n\ge1$. We have $\displaystyle{{x}_{2}}-\sqrt{a}=\frac{1}{2}\left( \frac{a}{{{x}_{1}}}+{{x}_{1}} \right)-\sqrt{a}=\frac{1}{2}\cdot \frac{{{({{x}_{1}}-\sqrt{a})}^{2}}}{{{x}_{1}}}>0,$ so this shows that $x_2>\sqrt a$. Suppose now that $x_n>\sqrt a,$ so $x_n>0$ and in the same fashion we have $\displaystyle{{x}_{n+1}}-\sqrt{a}=\frac{1}{2}\left( \frac{a}{{{x}_{1}}}+{{x}_{1}} \right)-\sqrt{a}=\frac{1}{2}\cdot \frac{{{({{x}_{n}}-\sqrt{a})}^{2}}}{{{x}_{n}}}>0,$ implying that $x_{n+1}>\sqrt a,$ so $x_n>\sqrt a$ for all $n\ge1.$ Let's now show that $x_{n+1}>x_n$ for all $n\ge1$. Since $0<\sqrt a<x_2$ we have $\displaystyle\frac{a}{{{x}_{2}}}-{{x}_{2}}=\frac{a-x_{2}^{2}}{{{x}_{2}}}<0\implies \frac{a}{{{x}_{2}}}+{{x}_{2}}<2{{x}_{2}}\implies {{x}_{3}}<{{x}_{2}}.$ In the same fashion for $0<\sqrt a<x_{n+1}$ we get $\displaystyle\frac{a}{{{x}_{n+1}}}-{{x}_{n+1}}=\frac{a-x_{n+1}^{2}}{{{x}_{n+1}}}<0\implies \frac{a}{{{x}_{n+1}}}+{{x}_{n+1}}<2{{x}_{n+1}},$ so ${{x}_{n+2}}<{{x}_{n+1}}.$

Finally $x_n>\sqrt a$ and $x_{n+1}<x_n,$ hence $x_n$ is a decreasing bounded below sequence, so it has a limit.

 

[alyafey22]

But we know that then the sequence converges to its infimum,or am I wrong?

Yes. A decreasing bounded below sequence converges to its infimum.