Convergence of a Recursive Sequence

In summary, the sequence {an} defined recursively by a1=1 and an+1=\frac{1}{2+a subn}, n\geq1 is convergent. This can be shown by observing that the sequence is bounded below and by checking a few terms to see that it is decreasing. However, induction does not work in this case. Another approach is to use the contraction mapping theorem by Banach. Alternatively, you can show that the "odd" and "even" elements of the sequence form decreasing and increasing subsequences, respectively.
  • #1
life is maths
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Homework Statement



A sequence {an} defined recursively by a1=1 and an+1=[itex]\frac{1}{2+a subn}[/itex], n[itex]\geq[/itex]1. Show that the sequence is convergent.

Homework Equations


If a sequence is bdd below and decreasing or it is bdd above and increasing, then it is convergent.

The Attempt at a Solution


{an}[itex]\geq[/itex]0, hence it is bounded below. I checked a few terms from the beginning and obtained a decreasing sequence. I tried induction to show this, but it didn't work. Base case is O.K. a0>a1 Suppose ak>ak+1.
Then we get
ak+2=[itex]\frac{1}{2+a sub(k+1)}[/itex]>[itex]\frac{1}{2+a subk}[/itex]=ak+1, and ak+2>ak+1. I feel like I'm doing a stupid mistake, and I can't understand, if this is true, why the induction does not work. I would be grateful if you could help me. Thanks for your time and effort :)
 
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  • #2
Induction doesn't always work since the sequence is not descreasing. If you check more terms then you see that.

Are you familiar with the contraction mapping theorem by Banach?
 
  • #3
micromass said:
Induction doesn't always work since the sequence is not descreasing. If you check more terms then you see that.

Are you familiar with the contraction mapping theorem by Banach?

Ah, really? So that was my mistake... I have no idea about contraction mapping theorem.
Thanks for enlightening me :biggrin:
Is there a more practical way to show a sequence is convergent? :confused:
 
  • #4
Can I use the limit definition? I see no way out... :(
 
  • #5
You may be able to show that the "odd" elements of the sequence form a decreasing subsequence and the "even" elements an increasing subsequence ...
 

FAQ: Convergence of a Recursive Sequence

What is convergence of a sequence?

Convergence of a sequence refers to the behavior of a sequence of numbers as its terms approach a certain limit. In simpler terms, it describes how the terms of a sequence get closer and closer to a particular value as the sequence progresses.

How do you determine if a sequence is convergent or divergent?

To determine if a sequence is convergent or divergent, you can use various tests such as the limit test, the ratio test, the root test, or the comparison test. These tests involve calculating the limit of the sequence and comparing it to a known value or using a comparison sequence to see how it behaves.

What is the difference between absolute and conditional convergence of a sequence?

Absolute convergence of a sequence means that the sum of the absolute values of its terms is convergent. On the other hand, conditional convergence means that the sum of the terms is convergent, but the sum of the absolute values of the terms is not. In simpler terms, a sequence is absolutely convergent if all its terms are positive, while a sequence is conditionally convergent if it alternates between positive and negative terms.

Can a divergent sequence have a finite limit?

No, a divergent sequence cannot have a finite limit. If a sequence has a finite limit, it means that its terms are getting closer and closer to a particular value. However, a divergent sequence does not approach any specific value, so it cannot have a finite limit.

What is the significance of convergence of a sequence in real life?

In real life, convergence of a sequence is used in many areas of science and mathematics, such as in calculus, physics, and statistics. It helps in understanding the behavior of systems and processes that involve a progression of values. For example, it can be used to analyze the growth of a population, the decay of radioactive materials, or the convergence of a series in financial investments.

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