# Finding convergence of a recursive sequence

1. Oct 23, 2012

### muzak

1. The problem statement, all variables and given/known data
$x_{n+1} = (x_{n} + 2)/(x_{n}+3), x_{0}= 3/4$

2. Relevant equations

3. The attempt at a solution
I've worked out a few of the numbers and got 3/4, 11/15, 41/56, 153/209, ...
It seems to be monotone and bounded below indicating it does converge I think. I need help figuring out what it converges to if it does. I've never really done convergence on a recursive sequence.

2. Oct 23, 2012

### LCKurtz

Once you know it converges, it is easy to find the limit. Suppose $x_n\to a$ and take the limit of both sides of the recursion to figure out $a$. Note this argument does not show it converges, only that if it does, the limit is $a$. You still have to show it is monotone and bounded (if it is).

Last edited: Oct 23, 2012
3. Oct 23, 2012

### phiiota

what is the first thing you think of when you see "n+1"?

Last edited: Oct 23, 2012
4. Oct 23, 2012

### HallsofIvy

Staff Emeritus
If the sequence converges, to x, say, then, taking the limit on both sides, we must have
$$x= \frac{x+ 2}{x+ 3}$$
Solve that for x.

Of course, it is not enough to say "It seems to be monotone and bounded below", you must show that it is monotone, you have to show it is.

5. Oct 23, 2012

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