SEL Series Expansion and Generalized Model Construction for the Real Number System via Series of Rationals

We present an algorithm for constructing infinite series expansion for real numbers, which yields generalized versions of three famous series expansions, namely, Sylvester series, Engel series, and Lüroth series expansions. Using series of rationals, a generalized model for the real number system is also constructed. 1. Introduction According to [1, 2], it is well known that each is uniquely representable as an infinite series expansion called Sylvester series expansion, which is of the form where Moreover, if and only if for all sufficiently large . An analogous representation (see [1–3]) also states that every real number has a unique representation as an infinite series expansion called Engel series expansion, which is of the form where Moreover, if and only if for all sufficiently large . For the last representation (see [1, 2]), it is also known that each is uniquely representable as an infinite series expansion called Lüroth series expansion, which is of the form where Moreover, if and only if is periodic. In 1988, A. Knopfmacher and J. Knopfmacher [4] further derived some elementary properties of the Engel series expansion and Sylvester series expansion and then developed two new methods for constructing new models for the real number system from the ordered field of rational numbers. These methods are partly similar to the one introduced by Rieger [5] for constructing the real numbers via continued fractions. In the present work, we will first introduce an algorithm for constructing an infinite series expansion for real numbers called Sylvester-Engel-Lüroth series expansion or SEL series expansion for short which yields generalized versions of three series expansions, namely, Sylvester series expansion, Engel series expansion, and Lüroth series expansion. Then we will establish some elementary properties of the SEL series expansion and develop a method for constructing a generalized model for the real number system using series of rationals, which yields generalized versions of Knopfmachers' models. 2. SEL Series Expansion Given any real number , write it as , where and . Then recursively define where is a positive rational number, which may depend on , for all . Using this algorithm and the same proof as in [1, 2], we have the following. Theorem 1. Let and assume that for all . Then is uniquely representable as an infinite series expansion called SEL series expansion, which is of the form where and for all . Lemma 2. Any series where converges to a real number such that . Furthermore, . By setting , and , for all in Theorem 1, and by setting ,