A generating function for a sequence {c_n} is a function f(x) with a power serie

Question

A generating function for a sequence {c_n} is a function f(x) with a power series representation: sigma n=0 to infinity (c_n)(x^n) where (c_(n+4))=(c_n) for all n>or=0.(a) Find the Maclaurin series (Taylor series centered at 0) for the function f(x)= x/(1-x-(x^2)). (b) Now let's find the Maclaurin series in another manner: Assume that f(x)= x/(1-x-(x^2))=sigma n=0 to infinity (c_n)(x^n) where (c_(n+4))=(c_n) for all n>or=0. Multiply both sides of the equation above by (1-x-(x^2)).Show that the coefficients (c_n) must satisfy the relationship: (c_(n+1)) = (c_n) + (c_(n-1)). (c) Explain why f(x) = x/(1-x-(x^2)) is the generating function for the Fibonacci sequence. Give a closed from expression for each member of the sequence.

Explanation / Answer

check it this will help u alot http://mathworld.wolfram.com/MaclaurinSeries.html http://mathworld.wolfram.com/TaylorSeries.html

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