Fibonacci Numbers

The Fibonacci numbers are given by the recurrence relation:

$F_{n} = F_{n-1} + F_{n-2}$

where $F_{0} = \{i|i=0, 1\}$ and $F_{1} = 1$ giving the possible sequences:

$0,1,1,2,3,5,8,13,21,34,55,89,144,\ldots$

or

$1,1,2,3,5,8,13,21,34,55,89,144,\ldots$

Implementation

Calculating Pi (π)

Fixed Ratio

Western: $\frac{22}{7} \approx \pi$ wrong at the thousandths.
Chinese: $\frac{355}{113} \approx \pi$ wrong at the ten-millionths.

Trigonometry

$\arctan(1) = \frac{\pi}{4}$

thus:

$\pi = 4 * \arctan(1)$

Newton

$\pi = 2 * \sum_{k=0}^{\infty} \frac{k!}{(2k +1)!!} = 2 * \sum_{k=0}^{\infty} \frac{2^{k}k!^2}{(2k+1)!} = 2 * (1 + \frac{1}{3} * (1 + \frac{2}{5} * (1 + \frac{3}{7} * (...)))$

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