Continued Fraction Expansion/Examples
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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:
\(\ds e\) | \(=\) | \(\ds \sqbrk {2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \ldots }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk {1, 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \ldots }\) |