Recursive sequence convergence

In summary, the conversation discusses how to prove the convergence of the sequence (x_n) given arbitrary real numbers x_1 and x_2. It is suggested to show that (x_n) is a Cauchy sequence and advice is given on how to get started. A proposed solution is to show that (x_n) is contractive and therefore convergent. The conversation ends with a question on how to find the limit of (x_n).
  • #1
antiemptyv
34
0

Homework Statement



Let [tex]x_1 < x_2[/tex] be arbitrary real numbers and let [tex]x_n :=\frac{1}{3}x_{n-1} + \frac{2}{3}x_{n-2}[/tex]. Prove the sequence [tex](x_n)[/tex] converges.

Homework Equations



Since this problem comes from the section on Cauchy sequences, I assume we will need to show [tex](x_n)[/tex] is a Cauchy sequence. I'm not so well-versed in working with the recursive sequences especially with arbitrary initial values.

Any advice on getting started?
 
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  • #2
would this be a valid solution? it looks like i can show the sequence is contractive.

[tex]|x_{n+1}-x_n| = |\frac{1}{3}x_n + 2 x_{n-1} - x_n | = \frac{2}{3}|x_{n-1} - x_n|[/tex]

Thus [tex](x_n)[/tex] is contractive, so it is convergent.
 
Last edited:
  • #3
OK. so now how do we go about finding the limit of [tex](x_n)[/tex]?
 

FAQ: Recursive sequence convergence

What is a recursive sequence?

A recursive sequence is a sequence of numbers where each term after the first is determined by a formula that uses one or more of the previous terms.

What is convergence in a recursive sequence?

Convergence refers to the behavior of a recursive sequence as the number of terms increases. A recursive sequence is said to converge if its terms eventually approach a fixed value as the number of terms increases.

How is convergence determined in a recursive sequence?

Convergence is determined by observing the behavior of the terms of a recursive sequence as the number of terms increases. If the terms approach a fixed value, the sequence is said to converge. If the terms do not approach a fixed value, the sequence is said to diverge.

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

Absolute convergence occurs when the sequence converges regardless of the starting value, while conditional convergence only occurs for certain starting values. In other words, absolute convergence means the sequence will converge no matter what, while conditional convergence means the sequence will only converge for certain starting values.

How can the convergence of a recursive sequence be proven?

There are various methods to prove the convergence of a recursive sequence, such as the ratio test, the root test, and the direct comparison test. These tests involve determining the behavior of the terms and comparing them to known convergent or divergent sequences.

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