Let x(n+1)= (3*x(n)) / (x(n))^2+2 n belongs in N, xo>=0 is some starting value
a) Show that if the sequence converges to a finite limit then this is 0 or 1.
b) Show that if 0<x(n)<1 then 0<x(n)<x(n+1)<1 and determine the limit behavior of the sequence when 0<x0<1.
c) Show that if 1<x(n)<2 then 1<x(n+1)<x(n) and determine the limit behavior of the sequence when 1<x0<2
d) What happens for x0>=2?
My work and if someone can check it and correct me if I am anywhere wrong.
a) If x(n) tends to K in R then so does x(n+1) so K=3K/K^2+2 so K=0 or K=1 or K=-1 but we cross out K=-1 since x0>=0 and so ...>x2>x1>x0 (x0>0) or all x(n)=0 (x0=0)
b) x(n+1)-x(n)=(x(n)*(1-(x(n))^2)) / ((x(n))^2+2) which is >0 if 0<x(n)<1 and so x(n+1)>x(n). So the sequence is an increasing one with 0<x(n)<x(n+1)<1, since the only positive finite limit is 1.
When 0<x0<1 our sequence is increasing and so 1>...>x2>x1>x0>0. It is bounded above by 1 since is the only positive finite limit.
c) If 1<x(n)<2 then x(n+1)-x(n)<0 so x(n)>x(n+1) so our sequence is decreasing but >1 since is the only positive finite limit and so 1<x(n+1)<x(n)
When 1<x0<2 then 1<.....<x2<x1<x0 our sequence is decreasing and bounded below by 1.
d) If x0=2 then all x(n) are zero so we have a sequence full of zeros.
If xo>2 we observe that x1<x0 but 1>...>x3>x2>x1 and our sequence is bounded above by 1 since is the only possible finite limit.
This is my work. Could anyone please check it and confirm my work? Thanks in advance, appreciate it!!
What do you mean? My approach is completely wrong? Why in b the recursive relation is 3xn/(x(n)^2+2n)?
Oh sorry, now I understand. I meant n belongs in N in my first sentence. So the sequence is just x(n+1)= (3*x(n)) / (x(n))^2+2
Thank you very much for the help :)
Sorry Chi Sigma for bothering again. I understood your logic. My solution was correct? I mean I prefer to work on my approach, so if you can tell me if it was correct or if I have to add anything I would greatly appreciate it! Thank you!
Thank you very much Chi Sigma, appreciate all your help :)
Hello Chi Sigma, sorry for bothering again. I want to ask if there is anything more to add to my answers for part b and c so I can deduce that my series is converging in that cases. I mean the pattern is obvious but is there something to add up to my answers to explain them better mathematically?