(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have a problem that states that A_{n}= [tex]\sqrt{A_{n-1}}[/tex]

, N is [tex]\geq[/tex] 1 where A_{0}is a positive constant.

I have to show that the sequence

{A_{n}} is decreasing if A_{0}> 1 and increasing if A_{0}< 1.

Show that the sequence is bounded below if A_{0}> 1 and bounded above

if A_{0}< 1.

Also I have to show the limit as n [tex]\rightarrow\infty A_n = 1[/tex]

3. The attempt at a solution

Now I'm pretty sure that I am right so I just need some verifcation here. This is what I did:

A_{0}= [tex]\sqrt{A_{1-1}}[/tex] = [tex]\sqrt{A_0}[/tex]

[tex]\sqrt{A_0}[/tex] < A_0[/tex]

A_{0}< (A_{0})^{2}

Call A_{0}= F(n)

F(n) > 1

F(n) = [tex]\sqrt{2}[/tex]. F(n) < N [[tex]\sqrt{2}[/tex] < 2]

F(n) < 1

F(n) = [tex]\sqrt{1/4}[/tex]. F(n) > N [[tex]\sqrt{1/4}[/tex] = 1/2 > 1/4]

So {A_{n}} decreases if A_{0}> 1 and increases if A_{0}< 1.

Because of the behavior of A_{n}in relation to N, A_{n}will always be less than N itself. If A_{n}> 1 and will be bounded below N.

Likewise if A_{n}A_{n}< 1 A_{n}will be greater than N and will always be bounded above.

3.

n [tex]\stackrel{Lim}{\rightarrow}[/tex][tex]\infty[/tex] |[tex]\sqrt{A_n/A{n-1}}[/tex] = 1

Via the ratio test.

So, is this a valid approach or am I off?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Solving a problem with Monotonic convergence.

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