Journal of Integer Sequences, Vol. 17 (2014), Article 14.4.4

On Error Sums for Square Roots of Positive Integers with Applications to Lucas and Pell Numbers

Carsten Elsner
Fachhochschule für die Wirtschaft
University of Applied Sciences
Freundallee 15
D-30173 Hannover


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.

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(Concerned with sequences A000032 A000045 A000129 A040968 A235861.)

Received September 9 2013; revised version received January 21 2014. Published in Journal of Integer Sequences, February 16 2014.

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