Series of Error Terms for Rational Approximations of Irrational Numbers

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.

Series of Error Terms for Rational Approximations of Irrational Numbers

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

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