Abstract

Let p_{n}/q_{n} be the n-th convergent of a real irrational number α, and let ε_{n} = αq_{n} − p_{n}. In this paper we investigate various sums of the type ∑_{m} ε_{m}, ∑_{m} |ε_{m}|, and ∑_{m} ε_{m}x^{m}. The main subject of the paper is bounds for these sums. In particular, we investigate the behaviour of such sums when α is a quadratic surd. The most significant properties of the error sums depend essentially on Fibonacci numbers or on related numbers.