# 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 $$f(\frac{1}{2})$$!

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?