Several types of infinite series are considered, which are defined by a fixed real number α and the denominators and numerators of the convergents of α. In this paper we restrict α to the irrational square roots of positive integers. We express the corresponding error sums in terms of a finite number of convergents. It is shown that an error sum formed by convergents with even indices takes only rational values. Two applications for error sums with α = √ 5 and α = √ 2 are given, where the convergents are composed of Lucas and Pell numbers, respectively.