Semiconvergent
= Semiconvergent
= Intermediate fraction
{synonym}
Given consecutive <continued fraction convergents> $p_{k-1}/q_{k-1}$ and $p_k/q_k$, the intermediate fractions $(tp_k+p_{k-1})/(tq_k+q_{k-1})$ for $1\le t\le a_{k+1}$ are semiconvergents, where $a_{k+1}$ is the next partial quotient. The endpoint is the next convergent. These fractions matter in <best rational approximation> when a <denominator> bound cuts between consecutive convergents.