Finding the limit of a recurrence equation.

In summary, the conversation is about a problem in a math journal where a sequence (a_n) is defined and a limit needs to be found in terms of a. The speaker is still working on solving it and is asking for any insights on recurrence equations or limits. They also mention a tip about finding the limit using a recurrence equation, but note that this particular recurrence is not in that form. They suggest using some trickery to put it in that form, potentially by considering a function of a for a fixed, large n.
  • #1
iironiic
9
0
I have been working on a problem proposed in a math journal, and there is only one thing I need to figure out. Here it is:

Let [itex](a_n)[/itex] be a sequence defined by [itex]a_1 = a[/itex] and [itex]a_{n+1} = 2^n-\sqrt{2^n(2^n-a_n)}[/itex] for all [itex]0 \leq a \leq 2[/itex] and [itex]n \geq 1[/itex]. Find [itex]\lim_{n \rightarrow \infty} 2^n a_n[/itex] in terms of [itex]a[/itex].

What I figured out so far:

Let [itex]A = \lim_{n \rightarrow \infty} 2^n a_n[/itex].

When [itex]a = 0[/itex], [itex]A = 0[/itex].

When [itex]a = \frac{1}{2}[/itex], [itex]A = \frac{\pi^2}{9}[/itex].

When [itex]a = 1[/itex], [itex]A = \frac{\pi^2}{4}[/itex].

When [itex]a = \frac{3}{2}[/itex], [itex]A = \frac{4\pi^2}{9}[/itex].

When [itex]a = 2[/itex], [itex]A = \pi^2[/itex].

I'm still trying to figure it out. Any insight on recurrence equations or limits would be greatly appreciated! Thanks!
 
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  • #2
Here's a tip: if the recurrence is [itex]a_{n+1}=f(a_n)[/itex] where f is continuous, then the limit is a solution to a=f(a). Intuitively, this is because since points near the limit change little, we reason that the limit ought to remain fixed under the recurrence.

Your recurrence is not in that form (since f depends on n), but some trickery may be used to put it there, for instance by considering f(a,n) as a function of a, for fixed, large n. (It'll be tough to justify rigorously, but it may be a good starting point.)
 

1. What is a recurrence equation?

A recurrence equation is a mathematical formula that defines a sequence of values by relating each value to one or more previous values in the sequence. It is often used to model the behavior of systems that change over time.

2. How do you find the limit of a recurrence equation?

To find the limit of a recurrence equation, you need to solve for the infinite term in the sequence. This can be done by repeatedly substituting the previous value into the equation until the sequence converges to a single value. This value is the limit of the recurrence equation.

3. What is the importance of finding the limit of a recurrence equation?

Finding the limit of a recurrence equation allows us to understand the long-term behavior of a system or process. It can also help us make predictions and make informed decisions based on the behavior of the system.

4. What are some common techniques for finding the limit of a recurrence equation?

Some common techniques for finding the limit of a recurrence equation include using algebraic manipulation, using a graphing calculator or software, and using mathematical induction to prove the limit exists.

5. Can the limit of a recurrence equation be infinite?

Yes, the limit of a recurrence equation can be infinite if the sequence grows without bound. This can occur when the values in the sequence increase at an increasing rate, leading to an unbounded limit.

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