haXadecimal
- 3
- 0
Convergence of oscillating sequence
Hi, I have to prove that an oscillating sequence converges, I am having some difficulty with the proof.
The sequence is [tex]c_{n+1} = \frac{1}{1+c_{n}} , c_{1} = 1[/tex]
So, I've calculated the first few terms and have seen that the sequence oscillates. I know that I need to prove:
1) The differences alternate in sign.
2) The absolute differences decrease.
3) The absolute differences approach 0.
I have proved 1, using:
[tex] c_{n+1} - c_{n} = \left(\frac{1}{1+c_{n}}\right) - \left(\frac{1}{1+c_{n-1}}\right)<br /> =\frac{1+c_{n-1}-1-c_{n}}{1+c_{n-1}+c_{n}+c_{n-1}c_{n}}[/tex]
[tex] =\frac{-(c_{n}-c_{n-1})}{1+c_{n-1}+c_{n}=c_{n-1}c_{n}}[/tex]
And since all terms are positive, the denomenator will be positive and the difference between two terms with alternate in sign from the difference between the previous two terms.
I now am having trouble proving 2 and 3. I'm not exactly sure what to do; the example in my book is not very helpful.So far I have:
[tex] |c_{n+1}-c_{n}| < |c_{n} - c_{n-1}|[/tex]
but that's not much... If anyone could help, that would be great! Thanks!
Hi, I have to prove that an oscillating sequence converges, I am having some difficulty with the proof.
The sequence is [tex]c_{n+1} = \frac{1}{1+c_{n}} , c_{1} = 1[/tex]
So, I've calculated the first few terms and have seen that the sequence oscillates. I know that I need to prove:
1) The differences alternate in sign.
2) The absolute differences decrease.
3) The absolute differences approach 0.
I have proved 1, using:
[tex] c_{n+1} - c_{n} = \left(\frac{1}{1+c_{n}}\right) - \left(\frac{1}{1+c_{n-1}}\right)<br /> =\frac{1+c_{n-1}-1-c_{n}}{1+c_{n-1}+c_{n}+c_{n-1}c_{n}}[/tex]
[tex] =\frac{-(c_{n}-c_{n-1})}{1+c_{n-1}+c_{n}=c_{n-1}c_{n}}[/tex]
And since all terms are positive, the denomenator will be positive and the difference between two terms with alternate in sign from the difference between the previous two terms.
I now am having trouble proving 2 and 3. I'm not exactly sure what to do; the example in my book is not very helpful.So far I have:
[tex] |c_{n+1}-c_{n}| < |c_{n} - c_{n-1}|[/tex]
but that's not much... If anyone could help, that would be great! Thanks!
Last edited:
)