Continued Fraction Expansion of Golden Mean

Theorem

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

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

The $n$th convergent is given by:

$C_n = \dfrac {F_{n + 1} } {F_n}$

where $F_n$ denotes the $n$th Fibonacci number.

This continued fraction expansion has the slowest rate of convergence of all simple infinite continued fractions.

Let:

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

Then:

$\ds x$

$=$

$\ds 1 + \frac 1 x$

substituting for $x$

$\ds \leadsto \ \ $

$\ds x^2$

$=$

$\ds x + 1$

$\ds \leadsto \ \ $

$\ds x^2 - x - 1$

$=$

$\ds 0$

$\blacksquare$

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): golden section
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): golden section