A question in prooving function convergence

1. Jan 22, 2008

transgalactic

2. Jan 22, 2008

HallsofIvy

Staff Emeritus
a0= 1, an+1= (an+ 1)/(an+ 2)

I take it you want to prove that it is decreasing. It's clearly bounded below (by 0), so it has a limit. And then find the limit. a1= (1+ 1)/(1+ 2)= 2/3< 1. That you have.

Now, suppose, for some k, ak> ak+1. Then ak+1+1= (ak+1+1)/ak+1+ 2). Again you have that but, as you say, since both numerator and denominator are larger than in ak+1, that doesn't tell you anything. Perhaps it would help to recognise that (x+ 1)/(x+ 2)= 1- 1/(x+ 2). If uk+1< uk then uk+1+ 2< uk+ 2 so 1/(uk+1+ 2)> 1/uk and then -1/(uk+1)< -1/uk.

You then solve t= (t+1)/(t+2) and get two solutions. Of course, only one of those is the limit of the sequence. The fact that only one of them is positive should make it clear which!

3. Jan 23, 2008

transgalactic

i tried to use what you told me
i have written your explanation several times
some steps in your post that you say "then"
i cant understand how you got them
and how i go further to proove my inequality

can you please wright me the solution to this problem
??

4. Jan 23, 2008

HallsofIvy

Staff Emeritus
No, I can't!