Lets1 = √(6),s2 = √(6+√(6)),s3 = √(6+√(6+√(6))), and in general definesn+1 = √(6+sn). Prove that (sn) converges, and find its limit.

*note: "√" means "the square root of"

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- Apr 8th 2009, 05:29 PMbearej50Monotone and Cauchy Sequences
Let

*s*1 = √(6),*s*2 = √(6+√(6)),*s*3 = √(6+√(6+√(6))), and in general define*s**n*+1 = √(6+*s**n*). Prove that (*s**n*) converges, and find its limit.

*note: "√" means "the square root of"

- Apr 8th 2009, 05:44 PMJhevon
show that the sequence is monotonic (increasing) and bounded. this will show that it has a limit (by what theorem?) you can prove both pretty easily by induction. do you see what to prove for each of these?

to find the limit, note that , call this limit .

then, since we have

you want the positive solution (why?) - Apr 8th 2009, 06:55 PMbearej50
I am still confused...

- Apr 8th 2009, 07:16 PMJhevon
- Apr 9th 2009, 07:11 AMbearej50
I know that I have to prove that s

*n*+1 > s*n*, which will prove that it is monotonically increasing. Then I would have to find an upper bound, U, for it and prove that s*n*< U and s*n*+1 < U. This would prove that it is bounded. Then, by the monotone convergence theorem, the sequence must be convergent and have a limit.

I am having trouble putting this all together in a proper proof. - Apr 9th 2009, 08:40 AMPlato
Here is the base case: .

Say it is true for K:

Done by induction: increasing and bounded. - Apr 11th 2009, 04:16 PMxalk
O.K.

Lets take the difference .................................................. ......................................1

Now if you multiply (1) by ,and do couple of calculations you get the formula:

.................................................. .....................................2

In this formula all the terms are +ve except the factor .

But since we can prove that for all n, then by using (2) we get .

Hence the sequence is increasing and bounded from above by 3.

So it is left to you to prove for all ,n and for all ,n.

Up to now we have proved :

a)**IF**the sequence converges ,then it must converge to 3 (because for all ,n)

b) The sequence converges.

BY (a) and (b) and using M.Ponens ,we have:

The limit of the sequence is 3.

Note usually the finding of a limit for this type of sequences consists of those two parts.

In the first part WE**Assume that the sequence converges**,and find its limit .

In the 2nd part we prove convergence