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
Zbl
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 
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/
@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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