Editing Simple continued fraction (section)

===Some useful theorems===
If <math>\ a_0\ ,</math> <math>a_1\ ,</math> <math>a_2\ ,</math> <math>\ \ldots\ </math> is an infinite sequence of positive integers, define the sequences <math>\ h_n\ </math> and <math>\ k_n\ </math> recursively:
{| border="0" cellpadding="5" cellspacing="10" align="none"
|-
|<math> h_{n} = a_n\ h_{n-1} + h_{n-2}\ ,</math>
|
|
|<math> h_{-1} = 1\ ,</math>
|
|<math> h_{-2} = 0\ ;</math>
|-
|<math> k_{n}= a_n\ k_{n-1} + k_{n-2}\ ,</math>
|
|
|<math> k_{-1} = 0\ ,</math>
|
|<math> k_{-2} = 1 ~.</math>
|}

<blockquote>'''Theorem 1.''' For any positive real number <math>\ x\ </math>

:<math> \left[\ a_0;\ a_1,\ \dots, a_{n-1}, x\ \right]=\frac{x\ h_{n-1} + h_{n-2}}{\ x\ k_{n-1} + k_{n-2}\ }, \quad \left[\ a_0;\ a_1,\ \dots, a_{n-1} + x\ \right]=\frac{h_{n-1} + xh_{n-2}}{\ k_{n-1} + x k_{n-2}\ }</math>
</blockquote>

<blockquote>'''Theorem 2.''' The convergents of <math>\ [\ a_0\ ;</math> <math>a_1\ ,</math> <math>a_2\ ,</math> <math>\ldots\ ]\ </math> are given by

:<math>\left[\ a_0;\ a_1,\ \dots, a_n\ \right] = \frac{h_n}{\ k_n\ } ~.</math>
or in matrix form,<math display="block">\begin{bmatrix}
h_n & h_{n-1} \\
k_n & k_{n-1}
\end{bmatrix} = \begin{bmatrix}
a_0 & 1 \\
1 & 0
\end{bmatrix} \cdots
\begin{bmatrix}
a_n & 1 \\
1 & 0
\end{bmatrix}</math>

'''Theorem 3.''' If the <math>\ n</math>th convergent to a continued fraction is <math>\ \frac{ h_n}{ k_n }\ ,</math> then
:<math> k_n\ h_{n-1} - k_{n-1}\ h_n = (-1)^n\ ,</math>
or equivalently
:<math> \frac{ h_n }{\ k_n\ } - \frac{ h_{n-1} }{\ k_{n-1}\ } = \frac{ (-1)^{n+1} }{\ k_{n-1}\ k_n\ } ~.</math>
</blockquote>

'''Corollary 1:''' Each convergent is in its lowest terms (for if <math>\ h_n\ </math> and <math>\ k_n\ </math> had a nontrivial common divisor it would divide <math>\ k_n\ h_{n-1} - k_{n-1}\ h_n\ ,</math> which is impossible).

'''Corollary 2:''' The difference between successive convergents is a fraction whose numerator is unity:

:<math> \frac{h_n}{k_n} - \frac{ h_{n-1} }{ k_{n-1} } = \frac{\ h_n\ k_{n-1} - k_n\ h_{n-1}\ }{\ k_n\ k_{n-1}\ } = \frac{ (-1)^{n+1} }{\ k_n\ k_{n-1}\ } ~.</math>

'''Corollary 3:''' The continued fraction is equivalent to a series of alternating terms:

:<math>a_0 + \sum_{n=0}^\infty \frac{ (-1)^n }{\ k_{n}\ k_{n+1}\ } ~.</math>

'''Corollary 4:''' The matrix
:<math>\begin{bmatrix}
h_n & h_{n-1} \\
k_n & k_{n-1}
\end{bmatrix} = \begin{bmatrix}
a_0 & 1 \\
1 & 0
\end{bmatrix} \cdots
\begin{bmatrix}
a_n & 1 \\
1 & 0
\end{bmatrix}</math>
has [[determinant]] <math>(-1)^{n+1}</math>, and thus belongs to the group of
<math>\ 2\times 2\ </math> [[unimodular matrix|unimodular matrices]] <math>\ \mathrm{GL}(2,\mathbb{Z}) ~.</math>

'''Corollary 5:''' The matrix<math display="block">\begin{bmatrix}
h_n & h_{n-2} \\
k_n & k_{n-2}
\end{bmatrix} = \begin{bmatrix}
h_{n-1} & h_{n-2} \\
k_{n-1} & k_{n-2}
\end{bmatrix}
\begin{bmatrix}
a_{n} & 0 \\
1 & 1
\end{bmatrix}</math>
has determinant <math>(-1)^na_n</math>, or equivalently,<math display="block"> \frac{ h_n }{\ k_n\ } - \frac{ h_{n-2} }{\ k_{n-2}\ } = \frac{ (-1)^{n} }{\ k_{n-2 }\ k_n\ }a_n</math>meaning that the odd terms monotonically decrease, while the even terms monotonically increase.

'''Corollary 6:''' The denominator sequence <math>k_0, k_1, k_2, \dots</math> satisfies the recurrence relation <math>k_{-1} = 0, k_0 = 1, k_n = k_{n-1}a_n + k_{n-2}</math>, and grows at least as fast as the [[Fibonacci sequence]], which itself grows like <math>O(\phi^n)</math> where <math>\phi= 1.618\dots</math> is the [[golden ratio]].

<blockquote>'''Theorem 4.''' Each (<math>\ s</math>th) convergent is nearer to a subsequent (<math>\ n</math>th) convergent than any preceding (<math>\ r</math>th) convergent is. In symbols, if the <math>\ n</math>th convergent is taken to be <math>\ \left[\ a_0;\ a_1,\ \ldots,\ a_n\ \right] = x_n\ ,</math> then
:<math> \left|\ x_r - x_n\ \right| > \left|\ x_s - x_n\ \right| </math>
for all <math>\ r < s < n ~.</math> </blockquote>

'''Corollary 1:''' The even convergents (before the <math>\ n</math>th) continually increase, but are always less than <math>\ x_n ~.</math>

'''Corollary 2:''' The odd convergents (before the <math>\ n</math>th) continually decrease, but are always greater than <math>\ x_n ~.</math>

<blockquote>'''Theorem 5.'''

:<math>\frac{1}{\ k_n\ (k_{n+1} + k_n)\ } < \left|\ x - \frac{ h_n }{\ k_n\ }\ \right| < \frac{1}{\ k_n\ k_{n+1}\ } ~.</math>
</blockquote>

'''Corollary 1:''' A convergent is nearer to the limit of the continued fraction than any fraction whose denominator is less than that of the convergent.

'''Corollary 2:''' A convergent obtained by terminating the continued fraction just before a large term is a close approximation to the limit of the continued fraction.<blockquote>'''Theorem 6:''' Consider the set of all open intervals with end-points <math>[0;a_1, \dots, a_n], [0;a_1, \dots, a_n+1]</math>. Denote it as <math>\mathcal C</math>. Any open subset of <math>[0, 1] \setminus \Q</math> is a disjoint union of sets from <math>\mathcal C</math>.</blockquote>'''Corollary:''' The infinite continued fraction provides a homeomorphism from the Baire space to <math>[0, 1] \setminus \Q</math>.