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May 11th, 2007, 08:07 AM   #1
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Power Series and Sequences!!!

Hi all!!

I am currently taking a Laplace Transforms and Fourier Series class, and have no idea where to start in this particular problem...

Use power series to find a formula for the sequence

a sub 0 = 1 a sub 1 = 1, a sub (n+2) - 4a sub (n+1) + 4a sub n = 0

Any help is much appreciated!!!!
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May 13th, 2007, 06:59 PM   #2
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This is a linear recurrence with constant coefficients, which you know how to solve explicitly (using the Kernel's lemma from linear algebra).

If you want to use power series (which seems to be asked here), you need to find a function y(t) such that y(0)=1, y'(0)=1 and y^(2n+2)=4y^(n+1)-4y^(n).Taking n=0 gives a necessary condition for y to exist, and determines such an y uniquely. The power series development of this solution y will be a candidate, but you'll still have to verify that it satisfies the required conditions.

I let you solve all of this numerically.
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May 14th, 2007, 02:37 AM   #3
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Have you learned about Binet's formula for the fibonacci sequence? Specifically about how it was derived? This would give you another way to give a formula in closed form for the recurrence.
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