Sum range of a quaternion series

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