Convergent Sequence of Square Roots

[gajohnson]

Homework Statement

Let $S_{1}=1$ and $S_{n+1}=\sqrt{2+S_n}$

Show that $\left\{S_n\right\}$ converges and find its limit.

Hint: First assume that the limit exists, then what is the possible value of the limit? Second, show that the sequence is increasing and bounded. Finally, follow the definition of convergence to show that the sequence converges.

Homework Equations

NA

The Attempt at a Solution

Well it is pretty clear that this converges to 2, so that's a start.

I am having difficulty constructing a good way to show that the sequence is increasing and bounded. Any help getting started would be nice.

Thanks!

 

[Dick]

Homework Statement

Let $S_{1}=1$ and $S_{n+1}=\sqrt{2+S_n}$

Show that $\left\{S_n\right\}$ converges and find its limit.

Hint: First assume that the limit exists, then what is the possible value of the limit? Second, show that the sequence is increasing and bounded. Finally, follow the definition of convergence to show that the sequence converges.

Homework Equations

NA

The Attempt at a Solution

Well it is pretty clear that this converges to 2, so that's a start.

I am having difficulty constructing a good way to show that the sequence is increasing and bounded. Any help getting started would be nice.

Thanks!

To show it's increasing you want to show sqrt(x+2)>x, right? For what range of x is that true? Try to solve the inequality.

 

[gajohnson]

To show it's increasing you want to show sqrt(x+2)>x, right? For what range of x is that true? Try to solve the inequality.

Well because $S_1=1$ is given, the sequence is strictly increasing for $x\in[1,2)$, and the sequence is monotonically increasing for $x\in[1,2]$.

Is showing this by solving the inequality enough to claim that the sequence is increasing and also bounded by 2 (since solving the above as an equality gives 2)?

 

[Dick]

Well because $S_1=1$ is given, the sequence is strictly increasing for $x\in[1,2)$, and the sequence is monotonically increasing for $x\in[1,2]$.

Is showing this by solving the inequality enough to claim that the sequence is increasing and also bounded by 2 (since solving the above as an equality gives 2)?

Yes, showing the inequality for the range x in [1,2) will show it. To show it's bounded you need to show the inequality sqrt(x+2)<2 holds in that range.

 

[gajohnson]

Yes, showing the inequality for the range x in [1,2) will show it. To show it's bounded you need to show the inequality sqrt(x+2)<2 holds in that range.

I believe I've got it now. Thanks for your help!