What is the limit of the recurrence relation for g_n?

In summary, the conversation discusses the sequence g_n which is defined by the recursive formula a_{n+1}=a_n^2-2 and g_n=\frac{a_1a_2...a_n}{a_{n+1}}. The question is to evaluate the limit of this sequence as n approaches infinity. Jason suggests that the limit is 1, but is looking for a nicer solution. However, it is mentioned that the limit may not exist in general and depends on the value of a_1. It is also noted that g_n either converges to zero or does not converge at all.
  • #1
Marian123
1
0
Hi all

Suppose that , a_{n+1}=a_n^2-2 and g_n=\frac{a_1a_2...a_n}{a_{n+1}}.
Evaluate \lim_{n\rightarrow \infty } g_n.

I have seen some information in this link. Besides, the sequence gn seems as a good rational approximation for \sqrt5. I know that the answer is 1, But I can't find any nice solution. Any hint is strongly appreciated.
 
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  • #2
I'm not sure the limit exists in general ...

If [tex]a_1=0[/tex] then [tex]g_n=0[/tex] for all n, so the limit is zero. If [tex]a_1=1[/tex], then g is an alternating sequence and the limit does not exist. If [tex]a_1=2[/tex], then [tex]g_n=2^{n-1}[/tex] which diverges. In fact, after playing for a few minutes, I can only get g to either converge to zero or not converge at all.

jason
 

1. What is a recurrence relation?

A recurrence relation is a mathematical equation that defines a sequence of numbers or objects by expressing each term in terms of one or more of the previous terms. It is often used to solve problems that involve repeating patterns or processes.

2. What are some examples of recurrence relations?

Some examples of recurrence relations include the Fibonacci sequence, the Towers of Hanoi problem, and the Tower of Brahma problem. These are all problems that involve a repeating pattern or process, and can be solved using a specific recurrence relation.

3. How is a recurrence relation different from a regular function?

A recurrence relation differs from a regular function in that it involves multiple steps or iterations to arrive at a solution. A regular function, on the other hand, can be evaluated in one step and does not require previous terms to be defined. Recurrence relations also often involve self-referential definitions, where the value of a term depends on previous terms in the sequence.

4. What are the practical applications of recurrence relations?

Recurrence relations have a wide range of practical applications in fields such as computer science, economics, and physics. They can be used to model and analyze complex systems, solve optimization problems, and even predict future trends or behaviors.

5. How do you solve a recurrence relation?

Solving a recurrence relation involves finding a closed-form solution, which is an explicit formula that can be used to calculate any term in the sequence without needing to refer to previous terms. This can be done through various methods such as substitution, generating functions, and divide and conquer algorithms.

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