# Proving convergence of recursive sequence.

• Darth Frodo
In summary, the problem is to prove convergence of the sequence {x_n} = \frac{1}{2}(x_{n-1} + \frac{c}{x_{n-1}}) for c>0. The student is struggling with recursive sequences and is seeking help. They consider using the theorem that a monotone increasing/decreasing and bounded sequence converges, and also suggest using the idea of letting x = g(x) to find the limit point. They then try to determine if the sequence is increasing or decreasing based on the value of c, but are not successful. Another student suggests looking at the graph of the function f(x) = \frac{1}{2}(x + \frac{c}{
Darth Frodo

## Homework Statement

Prove for c>0 the sequence ${x_n} = \frac{1}{2}(x_{n-1} + \frac{c}{x_{n-1}})$ converges.

## The Attempt at a Solution

This is proving difficult, I have never dealt with recursive sequences before. Any help would be appreciated. Thanks.

So, I have an idea, but I'm not sure if it proves convergence.

If $x_{n} = g(x_{n-1})$ then all I have to do it let $x = g(x)$.EDIT: Upon further thought, it does not as convergence => $x_{n} = g(x_{n-1})$. It is not a iff situation. Back at square one.

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You have the following handy theorem:

If ##(x_n)_n## is a monotone increasing/decreasing sequence and if the ##(x_n)_n## are bounded, then the ##(x_n)_n## converge.

I would start by trying to apply this one. Can you show the sequence is increasing/decreasing? Can you find a lower/upper bound?

Darth Frodo said:
So, I have an idea, but I'm not sure if it proves convergence.

If $x_{n} = g(x_{n-1})$ then all I have to do it let $x = g(x)$.

This is a nice remark, because if the sequence converges, you can find out the limit point this way. Indeed, you have ##x_n = g(x_{n-1})##. Thus

$$x = \lim_{n\rightarrow +\infty} x_n = \lim_{n\rightarrow +\infty} g(x_{n-1}) = g(x)$$

So can you figure out what the limit point ##x## is? (of course you still have to prove it actually converges!)

So this is what I have so far.

$x_0 = r$ r is a non-zero real number

$x_{n+1} = x_n(\frac{1}{2} + \frac{c}{2x_n^2})$Case 1: ##r^2 < c~ \Rightarrow## sequence is increasing

Case 2: ##r^2 > c ~\Rightarrow## sequence is decreasing

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Darth Frodo said:
So this is what I have so far.

$x_0 = r$ r is a non-zero real number

$x_{n+1} = x_n(\frac{1}{2} + \frac{c}{2x_n^2})$

Case 1: ##r^2 < c~ \Rightarrow## sequence is increasing

Case 2: ##r^2 > c ~\Rightarrow## sequence is decreasing

Not quite:

Suppose c = 4 and x_0 = 1, then x_1 = 5/2 and x_2 = 89/40, so neither increasing nor decreasing!

You're on the right track, though.

Darth Frodo said:

## Homework Statement

Prove for c>0 the sequence ${x_n} = \frac{1}{2}(x_{n-1} + \frac{c}{x_{n-1}})$ converges.

## The Attempt at a Solution

This is proving difficult, I have never dealt with recursive sequences before. Any help would be appreciated. Thanks.

To see what is happening, look at the graph of
$$y = f(x), \text{ where } f(x) = \frac{1}{2} \left( x + \frac{c}{x} \right)$$
Look up material about "cobweb plots"; see, eg., http://en.wikipedia.org/wiki/Cobweb_plot . This will give you some intuition about what is happening; it is not yet a 'proof', but may help get you started.

## 1. What is a recursive sequence?

A recursive sequence is a type of mathematical sequence where each term is defined by the previous term(s) in the sequence. In other words, the formula for finding a term in the sequence relies on the values of the previous term(s).

## 2. Why is it important to prove convergence of a recursive sequence?

Proving convergence of a recursive sequence is important because it allows us to determine if the sequence will eventually approach a single, fixed value, or if it will continue to oscillate or diverge. This information is crucial in understanding the long-term behavior of the sequence.

## 3. How do you prove convergence of a recursive sequence?

There are several methods for proving convergence of a recursive sequence, including the Monotone Convergence Theorem, the Squeeze Theorem, and the Cauchy Criterion. These techniques involve showing that the sequence is increasing, decreasing, or bounded, and that it approaches a limit as the number of terms increases.

## 4. What are some common mistakes when proving convergence of a recursive sequence?

One common mistake is assuming that a sequence is convergent without first verifying its properties and checking for potential divergence. Another mistake is using an incorrect formula or incorrectly identifying the limiting behavior of the sequence.

## 5. Can a recursive sequence converge to multiple values?

No, a recursive sequence can only converge to a single value. This is because the formula for finding each term is dependent on the previous term(s), so there can only be one limiting behavior as the number of terms increases.

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