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Convergence of a Sequence

  • Thread starter workerant
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[SOLVED] Convergence of a Sequence

1. 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 [Broken]



3. 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.
 
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Answers and Replies

tiny-tim
Science Advisor
Homework Helper
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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! :smile:

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? :smile:
 

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