Thread: Monotone sequences and Cauchy sequences

Suppose that x>0. Define a sequence (s_n) by s_1=k, and s_(n+1) = (((s_n)^2)+x)/(2s_n) for n elements in N. Prove that for any k>0, lim s_n = sqrt(x).

I've gotten that I need to show that (s_(n+1))^2)-x = (((s_n)^2)-x)^2/(4(s_n)^2)=>0, so that s_n=>sqrt(x) for n=>2.

Also I need to prove that (s_n) is decreasing for n=>2, by showing that (s_n)-(s_(n+1))>=0.

Unfortunately I don't know how to exactly show these two steps.

Any help would be great.

We will assume and so . Therefore, . We get from here . This was your first half of your problem.

Now, .
(The last inequality follows because )

Originally Posted by ThePerfectHacker

We will assume and so . Therefore, . We get from here . This was your first half of your problem.

Now, .
(The last inequality follows because )

We have :


.

AND

.

But . ,is not equal to
as your calculations show .

Please explain your calculations ,i am completely confused

Originally Posted by xalk

We have :


.

AND

.

But . ,is not equal to
as your calculations show .

Please explain your calculations ,i am completely confused

I fixed my post now, there was a mistake in it.