semiconvergent

semiconvergent

English

Etymology

semi- +‎ convergent

Noun

semiconvergent (plural semiconvergents)

1. (mathematics) A kind of fraction. If ${\\displaystyle {\\frac {h_{n-1}}{k_{n-1}}}}$, ${\\displaystyle {\\frac {h_{n}}{k_{n}}}}$ are successive convergents, then any fraction of the form ${\\displaystyle {\\frac {h_{n-1}+ah_{n}}{k_{n-1}+ak_{n}}}}$, where a is a nonnegative integer and the numerators and denominators are between the n and n+1 terms inclusive, is a semiconvergent.

   The semiconvergents to the continued fraction expansion of a real number include all the rational approximations which are better than any approximation with a smaller denominator.

Adjective

semiconvergent (not comparable)

1. (mathematics, of a matrix) For which the limit ${\\displaystyle \\lim _{k\\to \\infty }{\\mathbf {T} }^{k}}$ exists.