Show that two subsequences are monotonic and bounded

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Discussion Overview

The discussion revolves around the sequence $(a_{n})$ defined by the recurrence relation $a_{n+1}=1+\frac{2}{1+a_{n}}$. Participants are tasked with showing that the subsequences $a_{2k}$ and $a_{2k-1}$ are monotonic and bounded, and they also explore the limit of the sequence as $n$ approaches infinity. The discussion includes questions about whether to analyze the subsequences separately or if a broader approach could suffice.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest finding $a_{n+2}$ in terms of $a_{n}$ as a strategy to analyze the subsequences together.
  • One participant expresses confusion and requests further clarification on how to proceed with the analysis.
  • Another participant provides a formula for $a_{n+2}$, which differs from the previous claim, indicating a potential misunderstanding or error in calculations.
  • There is a discussion about comparing $a_{n+2}$ to $a_{n}$ using expressions like $a_{n+2}-a_{n}$ or $a_{n+2}/a_{n}$ to establish monotonicity and boundedness.
  • One participant calculates $a_{n+2}-a_{n}$ and presents the result as $\frac{3-a_{n}^{2}}{2+a_{n}}$.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct expression for $a_{n+2}$, as there are conflicting results presented. The discussion remains unresolved regarding the best approach to demonstrate the monotonicity and boundedness of the subsequences.

Contextual Notes

There are unresolved calculations and potential misunderstandings regarding the expressions for $a_{n+2}$. The discussion is contingent on the accuracy of these expressions and their implications for the subsequences.

evinda
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Hey again! :)
Let the sequence $(a_{n})$ with $a_{1}>0$ and $a_{n+1}=1+\frac{2}{1+a_{n}}$.Show that the subsequences $a_{2k}$ and $a_{2k-1}$ are monotonic and bounded.Find the limit $\lim_{n \to \infty} a_{n}$,if it exists.
Do I have to show separately that the two subsequences are monotonic and bounded??Or is there an other way to show it??Could I for example show that $a_{n}$ is monotonic and bounded??
 
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Re: show that two subsequences are monotonic and bounded

evinda said:
Hey again! :)
Let the sequence $(a_{n})$ with $a_{1}>0$ and $a_{n+1}=1+\frac{2}{1+a_{n}}$.Show that the subsequences $a_{2k}$ and $a_{2k-1}$ are monotonic and bounded.Find the limit $\lim_{n \to \infty} a_{n}$,if it exists.
Do I have to show separately that the two subsequences are monotonic and bounded?? Or is there an other way to show it?? Could I for example show that $a_{n}$ is monotonic and bounded??

They're probably hinting at a way to figure this out: find $a_{n+2}$ in terms of $a_{n}$. If you can reason from this expression adequately, you can kill both subsequences with one stone, to mix metaphors. The two subsequences they tell you to work with both have this in common: each term is two away from every other term in the original sequence.
 
Re: show that two subsequences are monotonic and bounded

Ackbach said:
They're probably hinting at a way to figure this out: find $a_{n+2}$ in terms of $a_{n}$. If you can reason from this expression adequately, you can kill both subsequences with one stone, to mix metaphors. The two subsequences they tell you to work with both have this in common: each term is two away from every other term in the original sequence.

I haven't understood.. (Worried) Could you explain it further to me??
 
Re: show that two subsequences are monotonic and bounded

evinda said:
I haven't understood.. (Worried) Could you explain it further to me??

Ok, let's do this one thing at a time. Can you find $a_{n+2}$ in terms only of $a_{n}$?
 
Re: show that two subsequences are monotonic and bounded

Ackbach said:
Ok, let's do this one thing at a time. Can you find $a_{n+2}$ in terms only of $a_{n}$?

I found: $a_{n+2}=1+\frac{1}{1+\frac{1}{a_{n}}}$ .How can I continue?
 
Re: show that two subsequences are monotonic and bounded

evinda said:
I found: $a_{n+2}=1+\frac{1}{1+\frac{1}{a_{n}}}$ .How can I continue?

Hmm. That's not what I get:
$$a_{n+2}= \frac{3+2a_{n}}{2+a_{n}}.$$
Can you show your working?
 
Re: show that two subsequences are monotonic and bounded

Ackbach said:
Hmm. That's not what I get:
$$a_{n+2}= \frac{3+2a_{n}}{2+a_{n}}.$$
Can you show your working?

I tried it again and found the same result.. :)
 
Re: show that two subsequences are monotonic and bounded

evinda said:
I tried it again and found the same result.. :)

Do you mean the same result as you got before, or the same result that I got?
 
Re: show that two subsequences are monotonic and bounded

Ackbach said:
Do you mean the same result as you got before, or the same result that I got?

The same that you get!
 
  • #10
Ah, so we're on the same page now. Can you compare $a_{n+2}$ to $a_{n}$ somehow? Maybe you can do $a_{n+2}-a_{n}$ or maybe $a_{n+2}/a_{n}$? If we can show this is monotonic and bounded, we'd be done with that part.
 
  • #11
Ackbach said:
Ah, so we're on the same page now. Can you compare $a_{n+2}$ to $a_{n}$ somehow? Maybe you can do $a_{n+2}-a_{n}$ or maybe $a_{n+2}/a_{n}$? If we can show this is monotonic and bounded, we'd be done with that part.

I found $a_{n+2}-a_{n}=\frac{3-a_{n}^{2}}{2+a_{n}}$.
 

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