Editing Simple continued fraction (section)

===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>