Editing Simple continued fraction (section)

==Semiconvergents==<!-- This section is linked from [[Complete quotient]] -->

If

:<math> \frac{h_{n-1}}{k_{n-1}},\frac{h_n}{k_n} </math>

are consecutive convergents, then any fractions of the form

: <math> \frac{h_{n-1} + mh_n}{k_{n-1} + mk_n},</math>

where <math>m</math> is an integer such that <math>0\leq m\leq a_{n+1}</math>, are called ''semiconvergents'', ''secondary convergents'', or ''intermediate fractions''. The <math>(m+1)</math>-st semiconvergent equals the [[Mediant (mathematics)|mediant]] of the <math>m</math>-th one and the convergent <math>\tfrac{h_n}{k_n}</math>. Sometimes the term is taken to mean that being a semiconvergent excludes the possibility of being a convergent (i.e., <math>0<m<a_{n+1}</math>), rather than that a convergent is a kind of semiconvergent.

It follows that semiconvergents represent a [[monotonic sequence]] of fractions between the convergents <math>\tfrac{h_{n-1}}{k_{n-1}}</math> (corresponding to <math>m=0</math>) and <math>\tfrac{h_{n+1}}{k_{n+1}}</math> (corresponding to <math>m=a_{n+1}</math>). The consecutive semiconvergents <math>\tfrac{a}{b}</math> and <math>\tfrac{c}{d}</math> satisfy the property <math>ad - bc = \pm 1</math>.

If a [[Diophantine approximation|rational approximation]] <math>\tfrac{p}{q}</math> to a real number <math>x</math> is such that the value <math>\left|x-\tfrac{p}{q}\right|</math> is smaller than that of any approximation with a smaller denominator, then <math>\tfrac{p}{q}</math> is a semiconvergent of the continued fraction expansion of <math>x</math>. The converse is not true, however.