Finding the limit of a recurrence equation.

iironiic

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:



I'm still trying to figure it out. Any insight on recurrence equations or limits would be greatly appreciated! Thanks!

alexfloo

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.)