The second set of questions is about the Fibonacci numbers. Recall that the Fibonacci sequence is a sequence of numbers fn defined by the fact that fi = 1, fa = 1, and for n 2 3, fn = fn-1 + fn-2- 1 a) Consider the rational function Find the first four terms of the Taylor Series centered at r = 0 of this function. Notice anything? 1 1b) Show that the Taylor Series centered at r= 0 of f(r) is precisely n-1 where fn is the Fibonacci Sequence. (Hint: Do not attempt this by doing calculus. Instead, set the series equal to f(1) and solve successively for the coefficients of that power series.) 1c) Note that -r-r+1 is a reducible polynomial over the reals with roots r= A. Use this to find a partial fraction decomposition for f(1). 1d) Use the partial fraction decomposition for f(r) to find an alternative power series representation for the function, and conclude that fn

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Chapter2: Second-order Linear Odes
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The second set of questions is about the Fibonacci numbers. Recall that the Fibonacci
sequence is a sequence of numbers fn defined by the fact that fi = 1, fa = 1, and for n 2 3,
fn = fn-1 + fn-2-
1 a) Consider the rational function
f(1) =
Find the first four terms of the Taylor Series centered at r = 0 of this function. Notice anything?
1
1b) Show that the Taylor Series centered at I= 0 of f(r) is precisely
Σ.
n-1
where fn is the Fibonacci Sequence. (Hint: Do not attempt this by doing calculus. Instead, set
the series equal to f(1) and solve successively for the coefficients of that power series.)
1c) Note that -r-r+1 is a reducible polynomial over the reals with roots r= .
Use this to find a partial fraction decomposition for f(1).
1d) Use the partial fraction decomposition for f(r) to find an alternative power series
representation for the function, and conclude that
fn
Transcribed Image Text:The second set of questions is about the Fibonacci numbers. Recall that the Fibonacci sequence is a sequence of numbers fn defined by the fact that fi = 1, fa = 1, and for n 2 3, fn = fn-1 + fn-2- 1 a) Consider the rational function f(1) = Find the first four terms of the Taylor Series centered at r = 0 of this function. Notice anything? 1 1b) Show that the Taylor Series centered at I= 0 of f(r) is precisely Σ. n-1 where fn is the Fibonacci Sequence. (Hint: Do not attempt this by doing calculus. Instead, set the series equal to f(1) and solve successively for the coefficients of that power series.) 1c) Note that -r-r+1 is a reducible polynomial over the reals with roots r= . Use this to find a partial fraction decomposition for f(1). 1d) Use the partial fraction decomposition for f(r) to find an alternative power series representation for the function, and conclude that fn
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