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.
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! ...
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.
We prove via induction that for all : .
Base of induction:
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.