Journal of Integer Sequences, Vol. 14 (2011), Article 11.1.4

Series of Error Terms for Rational Approximations of Irrational Numbers

Carsten Elsner
Fachhochschule für die Wirtschaft Hannover
Freundallee 15
D-30173 Hannover


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.

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(Concerned with sequences A000045 A001519 A007676 A007677 A041008 A041009.)

Received July 12 2010; revised version received January 10 2011. Published in Journal of Integer Sequences, January 28 2011.

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