Editing Champernowne constant (section)

== Continued fraction expansion ==
[[Image:Champernowne constant.svg|right|177x177px|thumb|The first 161 quotients of the continued fraction of the Champernowne constant. The 4th, 18th, 40th, and 101st are much bigger than 270, so do not appear on the graph.]]
[[Image:Champernowne constant logscale.svg|right|172x172px|thumb|The first 161 quotients of the continued fraction of the Champernowne constant on a [[logarithmic scale]].]]
The [[continued fraction|simple continued fraction]] expansion of Champernowne's constant does not [[continued fraction#Finite continued fractions|terminate]] (because the constant is not [[rational number|rational]]) and is [[continued fraction#Periodic continued fractions|aperiodic]] (because it is not an irreducible quadratic). A simple continued fraction is a continued fraction where the denominator is 1. The [[continued fraction|simple continued fraction]] expansion of Champernowne's constant exhibits extremely large terms appearing between many small ones. For example, in base 10,
: ''C''<sub>10</sub> = [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 4 57540 11139 10310 76483 64662 82429 56118 59960 39397 10457 55500 06620 04393 09026 26592 56314 93795 32077 47128 65631 38641 20937 55035 52094 60718 30899 84575 80146 98631 48833 59214 17830 10987, 6, 1, 1, ...]. {{OEIS|id=A030167}}

The large number at position&nbsp;18 has 166 digits, and the next very large term at position&nbsp;40 of the continued fraction has 2504 digits. That there are such large numbers as terms of the continued fraction expansion means that the convergents obtained by stopping before these large numbers provide an exceptionally good [[diophantine approximation|approximation]] of the Champernowne constant. For example, truncating just before the 4th partial quotient, gives 
<math display="block">10/81 = \sum_{k=1}^\infty k/10^k = 0.\overline{123456790},</math>
which matches the first term in the rapidly converging series expansion of the previous section and which approximates Champernowne's constant with an error of about {{math|1 × 10<sup>−9</sup>}}. Truncating just before the 18th partial quotient gives an approximation that matches the first two terms of the series, that is, the terms up to the term containing {{math|10<sup>−9</sup>}},
<math display="block">
\begin{align}
\frac{60499999499}{490050000000} &= 0.123456789+10^{-9}\sum_{k=10}^\infty k/10^{2(k-9)}=0.123456789+10^{-9}\frac{991}{9801}\\
 &= 0.123456789\overline{10111213141516171819\ldots90919293949596979900010203040506070809},
\end{align}
</math>
which approximates Champernowne's constant with error approximately {{math|9 × 10<sup>−190</sup>}}.

The first and second incrementally largest terms ("high-water marks") after the initial zero are 8 and 9, respectively, and occur at positions 1 and 2. Sikora (2012) noticed that the number of digits in the high-water marks starting with the fourth display an apparent pattern.<ref>Sikora, J. K. "On the High Water Mark Convergents of Champernowne's Constant in Base Ten." 3 Oct 2012. http://arxiv.org/abs/1210.1263</ref> Indeed, the high-water marks themselves grow doubly-exponentially, and the number of digits <math>d_n</math> in the ''n''th mark for <math>n\geqslant 3</math> are

:6, 166, <span style="color: red;">25</span>04, <span style="color: red;">33</span><span style="color: blue;">102</span>, <span style="color: red;">41</span><span style="color: rgb(0, 128, 0);">1</span><span style="color: blue;">100</span>, <span style="color: red;">49</span><span style="color: rgb(0, 128, 0);">11</span><span style="color: blue;">098</span>, <span style="color: red;">57</span><span style="color: rgb(0, 128, 0);">111</span><span style="color: blue;">096</span>, <span style="color: red;">65</span><span style="color: rgb(0, 128, 0);">1111</span><span style="color: blue;">094</span>, <span style="color: red;">73</span><span style="color: rgb(0, 128, 0);">11111</span><span style="color: blue;">092</span>, ...

whose pattern becomes obvious starting with the 6th high-water mark. The number of terms can be given by
<math display="block"> d_n = \frac{44 - 103 \times 2^{n-3} \times 5^{n-4}}{9} + \left(2^{n-1} \times 5^{n-4} \times n - 2n\right)
,n\in\mathbb{Z}\cap\left[3,\infty\right).</math>

However, it is still unknown as to whether or not there is a way to determine where the large terms (with at least 6 digits) occur, or their values. The high-water marks themselves are located at positions

:1, 2, 4, 18, 40, 162, 526, 1708, 4838, 13522, 34062, .... {{OEIS|id=A143533}}