English: Infinite geometric series sum, shown in linear complex-color (red is 1, chartreuse is i, cyan is -1, and indigo is -i), using the assumed-distributivity solution i.e. Σ_{j=0,1,2...∞}(z)^j ~ 1/(1-z) across the full complex plane of z values (with 1 located rightward of center and i located above center).
Regions inside the dashed unit-circle (where |z|≤1) are where the Taylor series converges, even though the sum appears to exist as finite everywhere except at the z=1 pole to the right of center. As you can see, in fact, this plot is nothing more than a plot of 1/z with the origin displaced to z=1.