Recursive sequences convergence

In summary, the sequence {a_n} defined as a_n+1 = a_n/[sqrt(0.5a_n + 1) + 1] converges to 0. This can be proven by comparing subsequent terms of the sequence and showing that the ratio between them is always less than 1/2, providing an upper bound that leads to the limit being 0. Directly applying the definition of a limit may not be successful in proving the convergence.
  • #1
ricardianequiva
14
0

Homework Statement



Let the sequence {a_n} defined by:

a_n+1 = a_n/[sqrt(0.5a_n + 1) + 1]

Prove that {a_n} converges to 0

Homework Equations


The Attempt at a Solution



I tried manipulating the equation but to no avail...
 
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  • #2
If the sequence approaches 0, then lim(n->infinity) a_n should be 0 as should a_n+1 correct?

If the series converges, then a_n and a_n+1 should both approach the limit L of the sequence as n becomes large.
 
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  • #3
yes. but i don't see how that helps
 
  • #4
ricardianequiva said:
yes. but i don't see how that helps

Ok let me try it a different way. Suppose instead of explicitly labeling the sequence as a sequence, I say that the sequence is actually a function f(x) which takes x_n and transforms it into x_n+1.

For this sequence, [itex]f(x)=\frac{x}{\sqrt{x/2+1}+1}[/itex]. If the sequence converges to a limit L, then clearly f(L)=L. Does that make more sense?
 
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  • #5
yes, i can see how that shows that the limit is 0, but i don't think its rigorous enough.
is there some way to actually show that the limit is 0, perhaps using definitions of limits?
 
  • #6
I had to delete my previous post because I found an error in it, but I think the argument can be made by comparison. Using the definition of a limit, for every [tex]\epsilon > 0[/tex], you want to prove there exists an N such that for [tex]n > N[/tex],[tex]|x_n - L| < \epsilon[/tex].

The question is vague enough and seems to ignore the lack of convergence if [tex]x_n = -2[/tex] so I'm going to restrict my argument to [tex]x_n > 0[/tex]. Clearly for arbitrary w and u > 0 such that w > u, then [tex]\frac{1}{\sqrt{0.5w+1}+1} < \frac{1}{\sqrt{0.5u+1}+1}[/tex]. Using this, you can place an upper bound on the ratio of subsequent terms of the sequence [tex]x_{n+1} = \frac{x_n}{\sqrt{0.5x_n+1}+1}[/tex] to say that [tex]x_{n+1} \leq \frac{1}{2} x_n[/tex], which is much easier to work with.

I don't think attempts to directly apply the definition of a limit without using this kind of workaround would be met with much success.
 
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1. What is a recursive sequence?

A recursive sequence is a sequence in which each term is defined by using the previous term in the sequence. In other words, the formula for each term refers back to the previous term. This creates a self-referencing pattern that continues indefinitely.

2. How is convergence defined for recursive sequences?

Convergence for recursive sequences is defined as the behavior of the sequence as the number of terms approaches infinity. If the terms of the sequence approach a specific value (known as the limit), then the sequence is said to converge. If the terms do not approach a specific value or approach different values, then the sequence is said to diverge.

3. What are some common methods for determining convergence of recursive sequences?

One common method for determining convergence is the ratio test, which compares the ratio of consecutive terms in the sequence to a value. If the ratio is less than 1, then the sequence is likely to converge. Another method is the root test, which compares the root of consecutive terms in the sequence to a value. If the root is less than 1, then the sequence is likely to converge.

4. Are there any special types of recursive sequences that always converge?

Yes, there are some special types of recursive sequences that always converge. One example is the geometric sequence, in which each term is multiplied by a constant factor. Another example is the arithmetic sequence, in which each term is added by a constant value. Both of these sequences have clear patterns and limits, making them easy to determine convergence.

5. What are some real-world applications of recursive sequences?

Recursive sequences have many real-world applications, particularly in the fields of mathematics and computer science. Some common examples include the Fibonacci sequence, which appears in nature and is also used in coding and encryption algorithms. Other examples include the recursive functions used in computer programming, such as the famous Towers of Hanoi problem.

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