Power & Taylor Series of Fibonacci
Interval of Convergence of a power series
a. Consider the Power series
sum of series from n=1 to infinity of FnX^n.
Use the ratio test to determine the open interval on which the pwr series converges.
b. Show that the Taylor series of the Fcn f(x) = x/(1-x-x^2) about x=0 is given by:
x/(1-x-x^2) = sum of series at n=1 to infinity of FnX^n,
where Fn is the Fibonnaci sequence.
Hint: CAll H(x) the sum of the series of FnXn on the interval of convergence found in part (a). i.e.,
set H(x) = sum of series at n=1 to infinity of FnX^n. By keeping in mind Fn+1=Fn + Fn-1 for n=2,3,4.., compute (1+x)H(x) and then find the value of x(1+x)H(x).
This shows how to use the ratio test to determine when the power series converges, and complete a proof regarding the Taylor series that involves the Fibonacci sequence.
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