Fibonacci numbers

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The Fibonacci sequence is defined by $$F_{n+2}=F_n+F_{n+1},F_1=F_2=1.$$

Properties

Theorem: The following series holds and converges for all $|x| \leq \dfrac{1}{\phi}$, where $\phi$ denotes the golden ratio: $$\dfrac{x}{1-x-x^2} = \displaystyle\sum_{k=1}^{\infty} F_k x^k.$$

Proof: proof goes here █

Videos

The Golden Ratio & Fibonacci Numbers: Fact versus Fiction
Doodling in Math: Spirals, Fibonacci, and Being a Plant (1 of 3)
Fibonacci mystery

See also

Golden ratio
Reciprocal Fibonacci constant
Lucas numbers

External links

The Fibonacci Quarterly
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