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May 9th, 2015, 05:14 AM   #1
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limit of a recursive sequence

let (a_n) be a sequence such that:

a_(n+2)=1/3*a_(n+1)+2/3*a_n

what is the limit of (a_n)?

[the fact it converges has already been proved earlier in the question and we are not given a(1) and a(2)]
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May 9th, 2015, 05:22 AM   #2
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The general solution is$$A\lambda_1^n + B\lambda_2^n$$ where $\lambda_1$ and $\lambda_2$ are roots of the characteristic equation $u^2 - \frac13u - \frac23=0$.

You probably don't need to know the constants $A$ and $B$.
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May 9th, 2015, 06:55 AM   #3
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how did u get this? (a derivation would be greatly appreciated but a brief explanation would also be very helpful if u dont mind)
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May 9th, 2015, 07:10 AM   #4
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I haven't seen a constructive proof, but a first order difference equation $a_{n+1} = ka_n$ has solution $a_0 k^n$ which could motivate us to seek solutions of the form $a_n = Ar^n$ to the second order equation $a_{n+2}+ba_{n+1}+ca_n=0$. Substituting that into the original equation and solving for $r$ gives the result that $r$ must be a root of the characteristic equation $u^2 + bu + c = 0$.

Since this equation has two roots, there are two independent solutions and any linear combination of them will also be a solution because the equation is linear.

One ought to prove that such a construction provides all solutions to the equation, but I've never seen such a proof.
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May 9th, 2015, 07:21 AM   #5
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the thing is this is from an analysis paper (the syllabus is a bit older but we have definitely not touched on difference equations yet, thats in another course altogether) so is there another way you could see to derive this using only 1st year analysis techniques?
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May 9th, 2015, 09:33 AM   #6
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It seems an odd question to give to someone who hasn't studied the subject at all!
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May 9th, 2015, 09:41 AM   #7
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Hello, fromage!

Quote:




























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