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

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


fuss/mathematics/sequences.txt · Last modified: 2017/02/22 18:30 (external edit)

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