Working with the Fibonacci sequence

Calculus Level 4

f(x)=n=0Fnxn\large f(x) = \sum_{n=0}^{\infty} F_{n}x^{n}

Define f(x)f(x) as the series above with FnF_{n} as the Fibonacci sequence with F0=0,F1=1F_{0} = 0, F_{1}=1. Calculate f(12)f(\frac{1}{2})!

If you think the answer diverges, submit your answer as -1000.

Bonus: Can you generalize the formula f(x)=n=0anxn\large \displaystyle f(x) = \sum_{n=0}^{\infty} a_{n}x^{n}, which the sequence an+2=an+1+an,a0=p,a1=qa_{n+2} = a_{n+1} + a_{n}, a_0 = p, a_1 = q? And what is the limit/boundaries of the generalization?


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