Abstract
We investigate some properties connected with the alternating Sylvester series and alternating Engel Series representation for real numbers, in terms of the integer digits involved. In particular, we look at an algorithm that leads to a general alternating series expansion for real numbers in terms of rationals and deduce the alternating Sylvester and alternating Engel series from this general series.KEY WORDS: alternating Sylvester series, alternating Engel series, rational numbers, alternating series expansion.
Highlights
The series of Engel and Sylvester (Galambos 1976)for representing real numbers have been studied in some detail
Theorem 2: The alternating Sylvester and alternating Engel series terminate after a finite number of terms if and only if A is rational
Note that for rational numbers with a finite expansion there is a possible ambiguity in the final term
Summary
The series of Engel and Sylvester (Galambos 1976)for representing real numbers have been studied in some detail. Much less known is the fact that there are alternating series representations of real numbers in terms of rationals corresponding to the above. In 1989 Knopfmacher and Knopfmacher introduced an algorithm according to which any real number may be expressed by a general alternating series of rationals. This algorithm is described [ ] [ ] below. O. Izevblzua Department of Mathematics University of Benin, Benin City. Theorem 2: The alternating Sylvester and alternating Engel series terminate after a finite number of terms if and only if A is rational.
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