Sum Range of a Quaternion Series

Abstract

In this paper, we obtain a result which implies, in particular, that for a quaternion z ∉ {−1, 1} with |z| = 1, the sum range of the series \( {\displaystyle \sum_n\frac{z^n}{n}} \) is a closed proper subfield of the division ring of quaternions ℍ isometrically isomorphic to the field of complex numbers ℂ.