Continued Fraction Expansion/Example

From ProofWiki
Jump to navigation Jump to search

Examples of Continued Fraction Expansions

Continued Fraction Expansion of Golden Mean

The golden mean has the simplest possible continued fraction expansion, namely $\sqbrk {1, 1, 1, 1, \ldots}$:

$\phi = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots} } }$


Continued Fraction Expansion of $\sqrt 2$

The continued fraction expansion of the square root of $2$ is given by:

$\sqrt 2 = \sqbrk {1, \sequence 2}$


Continued Fraction Expansion of $\pi$

The constant $\pi$ (pi) has the continued fraction expansion:

$\pi = \sqbrk {3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, \ldots}$


Continued Fraction Expansion of $e$

The constant Euler's number $e$ has the continued fraction expansion:

$e = \left[{2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, \ldots}\right]$