# Homework Help: Series Problem

1. Jul 31, 2007

### cybercrypt13

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

a1 = sqrt(3), a2 = sqrt( 3 + sqrt( 3) ), a3 = sqrt( 3 + sqrt( 3 + sqrt( 3) ) )

2. Relevant equations

Notice that each term is inside the sqrt of the previous term. I have no idea how to lay something like this out. Any help would be greatly appreciated.

3. The attempt at a solution

2. Jul 31, 2007

### daveb

What are you trying to do? Find an expression for the nth term? Find the sum of the series? What?

3. Jul 31, 2007

### cybercrypt13

sorry,

it says find a recursive formula for a n+1 in terms of an. and find its limit

4. Jul 31, 2007

### red_dog

$$a_{n+1}=\sqrt{3+a_n}$$

We have $$a_1<a_2$$
Suppose $$a_{n-1}<a_n$$.
Then $$a_n-a_{n+1}=\sqrt{3+a_{n-1}}-\sqrt{3+a_n}<0\Rightarrow a_n<a_{n+1}$$, so $$(a_n)_{n\geq 1}$$ is crescent.
We'll prove that $$\displaystyle a_n<\frac{1+\sqrt{13}}{2}$$
$$a_1<\sqrt{1+\sqrt{13}}{2}$$.
Suppose that $$a_n<\frac{1+\sqrt{13}}{2}$$.
Then $$a_{n+1}=\sqrt{3+a_n}<\sqrt{3+\frac{1+\sqrt{13}}{2}}=\frac{1+\sqrt{13}}{2}$$.
So the sequence is convergent. Let $$l=\lim_{n\to\infty}a_n$$.
Then $$l=\sqrt{3+l}\Rightarrow l=\frac{1+\sqrt{13}}{2}$$

5. Jul 31, 2007

### cybercrypt13

Thanks but I'm more confused now that I thought I was before. Where did your sqrt(13) come from?

thanks,

glenn

6. Jul 31, 2007

### cybercrypt13

Nevermind, I understand now. Thanks for the help...

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