Geodesic
Series of error terms for rational approximations of irrational numbers
Journal of integer sequences, Tome 14 (2011) no. 1.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 
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     author = {Elsner, Carsten},
     title = {Series of error terms for rational approximations of irrational numbers},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {14},
     number = {1},
     year = {2011},
     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/