The formulae for the convergents of continued fractions are always given recursively rather than in explicit form. This article derives explicit formulae for the convergents of the continued fraction expansions for square roots.

This note considers continuous functions f on an interval [a, b] and develops sufficient conditions to answer the question, for what points c in the interval is there a rectangle with height f(c) and some width beta-alpha that has a signed area equal to [integral of] beta over alpha f(x)dx?