only the two previous convergents are necessary. The initial "convergents" (required for the first two terms) are ⁰/₁ and ¹/₀. For example, here are the convergents for [0;1,5,2,2].

When using the Babylonian method to generate successive approximations to the square root of an integer, if one starts with the lowest integer as first approximant, the rationals generated all appear in the list of convergents for the continued fraction.