1.NounA kind of fraction. If
h
n
−
1
k
n
−
1
{\displaystyle {\frac {h_{n-1}}{k_{n-1}}}}
,
h
n
k
n
{\displaystyle {\frac {h_{n}}{k_{n}}}}
are successive convergents, then any fraction of the form
h
n
−
1
+
a
h
n
k
n
−
1
+
a
k
n
{\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."
2.AdjectiveFor which the limit
lim
k
→
∞
T
k
{\displaystyle \lim _{k\to \infty }{\mathbf {T}}^{k}}
exists.