Finding convergence of a recursive sequence

[muzak]

Homework Statement

$x_{n+1} = (x_{n} + 2)/(x_{n}+3), x_{0}= 3/4$

Homework Equations

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.

 

[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).

 

[phiiota]

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

 

[HallsofIvy]

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.

 

[hedipaldi]

Attached.