# The Fibonacci Pi Series

The purpose of this paper is both to observe, understand and appreciate the link between the Fibonacci sequence and the ubiquitous mathematical constant, $\pi$. It proves the following series for $\pi$, making use of the Fibonacci numbers.

$$\pi=\sum_{n=1}^{\infty}{\sum_{k=0}^{\infty}{\frac{4\cdot{}(-1)^k}{(2k+1)(F_{2n+1})^{2k+1}}}}$$
where $F_n$ is the nth Fibonacci number.
The derivation of this identity employs several different mathematical concepts.

Angle Sum Formulae
We begin with the well-known trigonometric sum formulae:
$$\sin(A+B)=\sin(A)\cdot{}\cos(B)+\cos(A)\cdot{}\sin(B)$$
$$\cos(A+B)=\cos(A)\cdot{}\cos(B)-\sin(A)\cdot{}\sin(B).$$
By dividing these expressions, we obtain a corresponding formula for the tangent:
$$\tan(A+B)=\frac{\sin(A+B)}{\cos(A+B)}=\frac{\tan(A)+\tan(B)}{1-\tan(A)\cdot{}\tan(B)}$$