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For the characteristic equation $a_2 \lambda^2 + a_1 \lambda + a_0 = 0$ of the difference equation $a_2 x_{n+2} + a_1 x_{n+1} + a_0 x_n = 0$, I remember there is a way to indicate if the function of $f(n) = x_n$ is in/decreasing or oscillating? Thank you~

Gerry Myerson
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1LiterTears
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1 Answers1

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You will have $$f(n)=Ar^n+Bs^n$$ where $A$ and $B$ depend on $x_1$ and $x_2$, and where $r$ and $s$ are the roots of the characteristic equation. That should enable you to determine increasing, decreasing, and alternating.

Strictly speaking, I've left out the case of repeated roots. I can get back to that, if you want.

Gerry Myerson
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  • Dear Gerry - thanks. Yes I can reach your conclusion so far, but I am confused about how to determine increasing/decreasing or alternating - including the case with repeated roots? Thanks so much! – 1LiterTears Feb 04 '14 at 06:54
  • Well, if $A$ and $B$ are positive, and $r\gt1\ge s\ge-1$, then $f(n)$ is (eventually) increasing (and unbounded). You have to look at a lot of cases like this, depending on the signs of $A$ and $B$, on whether $r$ and $s$ are real, on whether their absolute values exceed 1, and so on. Take it as an exercise to go through all the possibilities. – Gerry Myerson Feb 04 '14 at 06:58
  • Yes, that's exactly what I am looking for. Could you show me one example on how to determine the behavior of the function - for one like when $r>1 \geq 1 s \geq -1$, how can I decide it is increasing? – 1LiterTears Feb 04 '14 at 06:59
  • Why don't you try some experiments with various values of $r$ and $s$ (and $A$ and $B$) and see for yourself? – Gerry Myerson Feb 04 '14 at 07:02
  • I see - I thought there should be some canonical way to enumerate. – 1LiterTears Feb 04 '14 at 07:05
  • If there is, I'm sure you'll find it by doing the experiments. – Gerry Myerson Feb 04 '14 at 07:40
  • Good point, thanks Gerry. I'll put it up if I find it. – 1LiterTears Feb 04 '14 at 07:45