Working with the Fibonacci sequence

Calculus Level 4

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

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

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

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


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