Convergence of a Sequence

In summary, the conversation discusses a recursively defined sequence and how to prove its convergence and find its limit. The first step in the solution is to show that the sequence is increasing and bounded, which proves its convergence. The next step involves finding the first five terms of the sequence and then using this to prove the next step. Finally, the conversation provides a hint for how to go about proving the increasing and bounded nature of the sequence.
  • #1
workerant
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[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.
 
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  • #2
Welcome to PF!

workerant said:
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:
 

What is the definition of convergence of a sequence?

Convergence of a sequence is a mathematical concept that refers to the behavior of a sequence of numbers as its terms approach a specific value or limit. It is a fundamental concept in analysis and is used to determine the long-term behavior of a sequence.

How is convergence of a sequence related to limits?

The convergence of a sequence is closely related to the concept of limits. In fact, a sequence is said to converge to a specific limit if the terms of the sequence get closer and closer to that limit as the sequence progresses.

What does it mean for a sequence to diverge?

A sequence is said to diverge if the terms of the sequence do not approach a specific value or limit. This can happen if the terms of the sequence become increasingly larger or smaller, or if the terms oscillate between different values without approaching a specific limit.

What is the difference between pointwise and uniform convergence?

Pointwise convergence refers to the behavior of a sequence at individual points, while uniform convergence refers to the behavior of the sequence as a whole. In uniform convergence, the rate of convergence remains the same at every point, while in pointwise convergence, the rate of convergence can vary at different points.

How is the convergence of a sequence determined?

The convergence of a sequence can be determined by analyzing the behavior of its terms as the sequence progresses. Some common techniques used to determine convergence include the comparison test, the ratio test, and the root test, among others. Additionally, the behavior of the sequence can also be visualized using graphs or tables of values.

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