If |z| ≥ 1, the series diverges, since the terms do not approach 0. Otherwise consider the limit of the partial sums tj = zj+1 - 1 over z-1. The denominator is a constant, a scaling factor, so we only need ask whether the numerator converges. Since z is inside the unit circle, zj approaches the origin, and the numerator approaches -1. The limit of the series is -1/(z-1), or 1/(1-z).
As an example, consider 1 + 1/3 + 1/9 + 1/27 + 1/81 + … Here z = 1/3, hence the limit is the reciprocal of 2/3, or 3/2. If the series began with 1/9, omiting the first two terms, the limit would be 1/6.
If z is complex, replace z with its norm |z|. This produces a geometric sum of positive numbers that converges, hence it converges absolutely. Since the norms converge absolutely, the original series zj converges absolutely.