s_{0}=1.0, s_{i+1}=0.5*s_{i}

Center:0.0+0.0i; Zoom =0.5

Yesterday’s sequence reset after 30 steps. Without a reset, arithmetic sequences head off to infinity. Fractal pictures with them just have some ghostly shadows, no chaos. The first few orbit steps for small numbers bounce around, but soon everything heads off to infinity. Only light shadows from the early iteration remain.

The same holds for geometric sequences that go to infinity. But not all geometric sequences blow up. Here is one that converges to 0. Usually the fractals built on these sequences, such as this one, have a large capture set, and no need to reset after a few steps.