Geodesic
Series of error terms for rational approximations of irrational numbers
Journal of integer sequences, Tome 14 (2011) no. 1
Let $p_n/q_n $ be the $n$-th convergent of a real irrational number $\alpha $, and let $\varepsilon_n = \alpha q_n-p_n $. In this paper we investigate various sums of the type $\sum_{m} \varepsilon_m \), \(\sum_{m} \vert\varepsilon_m\vert $, and $\sum_{m} \varepsilon_m x^m $. The main subject of the paper is bounds for these sums. In particular, we investigate the behaviour of such sums when $\alpha $ is a quadratic surd. The most significant properties of the error sums depend essentially on Fibonacci numbers or on related numbers.
Classification : 11J04, 11J70, 11B39
Keywords: continued fractions, convergents, approximation of real numbers, error terms 
@article{JIS_2011__14_1_a1,
     author = {Elsner,  Carsten},
     title = {Series of error terms for rational approximations of irrational numbers},
     journal = {Journal of integer sequences},
     year = {2011},
     volume = {14},
     number = {1},
     zbl = {1248.11049},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2011__14_1_a1/}
}
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Elsner,  Carsten. Series of error terms for rational approximations of irrational numbers. Journal of integer sequences, Tome 14 (2011) no. 1. http://geodesic.mathdoc.fr/item/JIS_2011__14_1_a1/