Semiconvergent

Given consecutive continued fraction convergents $p_{k-1}/q_{k-1}$ and $p_k/q_k$, the intermediate fractions $(t{p}_k+p_{k-1})/(t{q}_k+q_{k-1})$ 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.