(adsbygoogle = window.adsbygoogle || []).push({}); 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)

=\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!

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# Homework Help: Converrgence of oscillating sequence

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