Geodesic
On error sums for square roots of positive integers with applications to Lucas and Pell numbers
Journal of integer sequences, Tome 17 (2014) no. 4
Several types of infinite series are considered, which are defined by a fixed real number $\alpha $ and the denominators and numerators of the convergents of $\alpha $. In this paper we restrict $\alpha $ 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 $\alpha = \sqrt 5$ and $\alpha = \sqrt 2$ are given, where the convergents are composed of Lucas and Pell numbers, respectively.
Classification : 11J70, 11J04, 11D09, 11B39
Keywords: error sum, continued fraction, convergent, pell's equation, Lucas number, pell number 
@article{JIS_2014__17_4_a6,
     author = {Elsner,  Carsten},
     title = {On error sums for square roots of positive integers with applications to {Lucas} and {Pell} numbers},
     journal = {Journal of integer sequences},
     year = {2014},
     volume = {17},
     number = {4},
     zbl = {1291.11099},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2014__17_4_a6/}
}
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%T On error sums for square roots of positive integers with applications to Lucas and Pell numbers
%J Journal of integer sequences
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Elsner,  Carsten. On error sums for square roots of positive integers with applications to Lucas and Pell numbers. Journal of integer sequences, Tome 17 (2014) no. 4. http://geodesic.mathdoc.fr/item/JIS_2014__17_4_a6/