Editing Simple continued fraction (section)

==Other continued fraction expansions==

===Periodic continued fractions===
{{main|Periodic continued fraction}}
The numbers with periodic continued fraction expansion are precisely the [[quadratic irrational|irrational solutions]] of [[quadratic equation]]s with rational coefficients; rational solutions have finite continued fraction expansions as previously stated. The simplest examples are the [[golden ratio]] φ = [1;1,1,1,1,1,...] and {{sqrt|2}} = [1;2,2,2,2,...], while {{sqrt|14}} = [3;1,2,1,6,1,2,1,6...] and {{sqrt|42}} = [6;2,12,2,12,2,12...]. All irrational square roots of integers have a special form for the period; a symmetrical string, like the empty string (for {{sqrt|2}}) or 1,2,1 (for {{sqrt|14}}), followed by the double of the leading integer.

===A property of the golden ratio φ===
Because the continued fraction expansion for [[golden ratio|φ]] doesn't use any integers greater than 1, φ is one of the most "difficult" real numbers to approximate with rational numbers. [[Hurwitz's theorem (number theory)|Hurwitz's theorem]]{{sfn|Hardy|Wright|2008|loc=Theorem 193}} states that any irrational number {{mvar|k}} can be approximated by infinitely many rational {{sfrac|''m''|''n''}} with

:<math>\left| k - {m \over n}\right| < {1 \over n^2 \sqrt 5}.</math>

While virtually all real numbers {{mvar|k}} will eventually have infinitely many convergents {{sfrac|''m''|''n''}} whose distance from {{mvar|k}} is significantly smaller than this limit, the convergents for φ (i.e., the numbers {{sfrac|5|3}}, {{sfrac|8|5}}, {{sfrac|13|8}}, {{sfrac|21|13}}, etc.) consistently "toe the boundary", keeping a distance of almost exactly <math>{\scriptstyle{1 \over n^2 \sqrt 5}}</math> away from φ, thus never producing an approximation nearly as impressive as, for example, [[Milü|{{sfrac|355|113}}]] for [[pi|{{pi}}]]. It can also be shown that every real number of the form {{sfrac|''a'' + ''b''φ|''c'' + ''d''φ}}, where {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, and {{mvar|d}} are integers such that {{math|1=''a'' ''d'' − ''b'' ''c'' = ±1}}, shares this property with the golden ratio φ; and that all other real numbers can be more closely approximated.

===Regular patterns in continued fractions===
While there is no discernible pattern in the simple continued fraction expansion of {{pi}}, there is one for {{math|''e''}}, the [[e (mathematical constant)|base of the natural logarithm]]:

:<math>e = e^1 = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, \dots],</math>

which is a special case of this general expression for positive integer {{mvar|n}}:

:<math>e^{1/n} = [1; n-1, 1, 1, 3n-1, 1, 1, 5n-1, 1, 1, 7n-1, 1, 1, \dots] \,\!.</math>

Another, more complex pattern appears in this continued fraction expansion for positive odd {{mvar|n}}:

:<math>e^{2/n} = \left[1; \frac{n-1}{2}, 6n, \frac{5n-1}{2}, 1, 1, \frac{7n-1}{2}, 18n, \frac{11n-1}{2}, 1, 1, \frac{13n-1}{2}, 30n, \frac{17n-1}{2}, 1, 1, \dots \right] \,\!,</math>

with a special case for {{math|1=''n'' = 1}}:

:<math>e^2 = [7; 2, 1, 1, 3, 18, 5, 1, 1, 6, 30, 8, 1, 1, 9, 42, 11, 1, 1, 12, 54, 14, 1, 1 \dots, 3k, 12k+6, 3k+2, 1, 1 \dots] \,\!.</math>

Other continued fractions of this sort are

:<math>\tanh(1/n) = [0; n, 3n, 5n, 7n, 9n, 11n, 13n, 15n, 17n, 19n, \dots] </math>

where {{mvar|n}} is a positive integer; also, for integer {{mvar|n}}:

:<math>\tan(1/n) = [0; n-1, 1, 3n-2, 1, 5n-2, 1, 7n-2, 1, 9n-2, 1, \dots]\,\!,</math>

with a special case for {{math|1=''n'' = 1}}:

:<math>\tan(1) = [1; 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, \dots]\,\!.</math>

If {{math|''I''<sub>''n''</sub>(''x'')}} is the modified, or hyperbolic, [[Bessel function]] of the first kind, we may define a function on the rationals {{sfrac|''p''|''q''}} by

:<math>S(p/q) = \frac{I_{p/q}(2/q)}{I_{1+p/q}(2/q)},</math>

which is defined for all rational numbers, with {{mvar|p}} and {{mvar|q}} in lowest terms. Then for all nonnegative rationals, we have

:<math>S(p/q) = [p+q; p+2q, p+3q, p+4q, \dots],</math>

with similar formulas for negative rationals; in particular we have

:<math>S(0) = S(0/1) = [1; 2, 3, 4, 5, 6, 7, \dots].</math>

Many of the formulas can be proved using [[Gauss's continued fraction]].

===Typical continued fractions===
Most irrational numbers do not have any periodic or regular behavior in their continued fraction expansion. Nevertheless, for [[almost all]] numbers on the unit interval, they have the same limit behavior.

The arithmetic average diverges: <math>\lim_{n\to\infty}\frac 1n \sum_{k=1}^n a_k = +\infty</math>, and so the coefficients grow arbitrarily large: <math>\limsup_n a_n = +\infty</math>. In particular, this implies that almost all numbers are well-approximable, in the sense that<math display="block">\liminf_{n\to\infty}  \left|
x - \frac{p_n}{q_n}
\right| q_n^2 = 0</math>[[Aleksandr Khinchin|Khinchin]] proved that the [[geometric mean]] of {{math|''a''<sub>''i''</sub>}} tends to a constant (known as [[Khinchin's constant]]):<math display="block">\lim_{n \rightarrow \infty } \left( a_1 a_2 ... a_n \right) ^{1/n} =
K_0 = 2.6854520010\dots</math>[[Paul Lévy (mathematician)|Paul Lévy]] proved that the {{mvar|n}}th root of the denominator of the {{mvar|n}}th convergent converges to [[Lévy's constant]]
<math display="block">\lim_{n \rightarrow \infty } q_n^{1/n} =
e^{\pi^2/(12\ln2)} = 3.2758\ldots</math>[[Lochs' theorem]] states that the convergents converge exponentially at the rate of<math display="block">\lim_{n\to\infty}\frac 1n \ln\left|
x - \frac{p_n}{q_n}
\right| = -\frac{\pi^2}{6\ln 2} </math>