Please reread you sequence definition.
AS posted, that sequence is not bounded.
So it must be wrong.
Hi, I had difficulty solving this problem. Actually, I'm stuck.
Suppose (Xn) in R is defined recursively by X1= 1 and , n= 1, 2, ...
Show (Xn) converges and find its limit.
And there's a hint: Notice that | Xn+2 - X n+1| < 1/9 |Xn - Xn+1|. Show (Xn) is Cauchy.
Thanks for any help.
I edited my mistake. Sorry again.
Why not learn to post in symbols? You can use LaTeX tags
[tex] \displaystyle x_{n+1}=\dfrac{1}{3+x_n} [/tex] gives
At least learn to use grouping symbols.
That wasted time and effort.
In terms of difference equation the sequence is the solution of the recursive equation...
(1)
The function is represented here...
It exists one attractive fixed point in and for any the sequence will converge to . An interesting detail: because is the sequence will be 'oscillating'. That is confirmed for ...
For the behavior of the sequence is stiil to be analysed...
A very interesting question! ...
Kind regards
In this post I will try to prove something more general.
Let be a sequence. Suppose there exist so that for all . I will prove that converges.
Proof:
We prove via induction that for all : .
Base of induction:
Of course:
Step of induction:
Let be such that: , hence:
.
...hence the statement is true for all .
Now, for all that we have:
Hence, is Cauchy sequence, and it converges.