Does This Recursively Defined Sequence Converge?

[workerant]

[SOLVED] Convergence of a Sequence

Homework Statement

Consider the following "recursively defined" sequence:

a1=0.3
a(n+1)=sqrt (an+1)

Compute the first first five terms and prove that it converges. Then, find the limit of the sequence.


Please see:

Problem 3, particularly parts c and d here for the complete problem:

http://rutcor.rutgers.edu/~ngoldberg/math152/ws0837.pdf

The Attempt at a Solution

I would try to show that the sequence is increasing and bounded because that shows that it converges. I am not sure, however, how to go about doing this.

I think I am okay with finding the first five terms. I suppose the second term would be a2= sqrt (a1+1)=sqrt (1.3) and so on and so forth and it would get quite complicated.

I am not sure how to go about the proof and finding the limit.


Thank you.

 

[tiny-tim]

Welcome to PF!

I would try to show that the sequence is increasing and bounded because that shows that it converges. I am not sure, however, how to go about doing this.

Hi workerant! Welcome to PF!

Do one step at a time!

First step is to prove that a_n+1 > a_n.

In other words, that √(1 + a_n) > a_n

Hint: what is the condition for √(1 + a) > a?