I am trying to figure out how can i get limit of recursive sequence.

For example this one.

Is there any common way of solving all recursive limits?

Printable View

- Jan 25th 2010, 04:14 AMPJaniLimits of recursive sequences
I am trying to figure out how can i get limit of recursive sequence.

For example this one.

Is there any common way of solving all recursive limits? - Jan 25th 2010, 05:05 AMHallsofIvy
IF the limit exists, then take the limit of both sides of the recursion equation: . If the limit exists and , then both of those limits are a. You get the equation . Solve that equation for a.

Of course, that only works**if**the sequence converges. Here you can see that the first few numbers are 1, 3/2, 7/4, .... You should be able to prove, by induction, that this sequence is increasing and has 2 as an upper bound. By the "monotone sequence theorem", then, the sequence converges. - Jan 25th 2010, 05:07 AMchisigma
The sequence is defined by the recursive equation...

(1)

... and from (1) we derive that...

(2)

The sequence admits finite limit if and only if , so that from (2) we derive...

(3)

Kind regards

- Jan 25th 2010, 05:40 AMPJani
HallsofIvy and chisigma:

thank you two

I know how to make induction on explicit sequences but i don't know how to make induction on recursive equations. - Jan 25th 2010, 05:55 AMHallsofIvy
Recursive sequences, because they give an explicit formula for in terms of , are easier to use with induction!

For example, to prove this sequence is increasing:

1) If n= 1, and so .

2) Assume that, for some k, . Then .

(I have used the fact that, since , .)

Its just as easy to prove that 2 is an upper bound for the sequence.