E1Reset. With a ={0.5, 0} the partial sum series converges to 2.

Shift a along the real axis and compare results with the simulation of the real geometric series.

E2: Choose a close to {1, 0}

The condition of convergence of the partial sum series is obviousely abs(a ) < 1.

What is the condition for the elements of the sequence?

E3: Choose a around {-1, 0}. Observe both charts.

E4:  Choose abs(a) < 1 with a small imaginary part and watch the behavior of the series.

E5: Increase the imaginary part and reflect what determines the character of the sequence spiral:

multiplication by a for each member increases the angle to the x-axes by the angle of a: arctg [imaginarypart(a)/realpart(a)]. Soon the points will be on multiple Riemann sheets.

E6: Look for a, where the spirals form straight radial arms, and analyze the cause of these symmetries.