In analogy to the real series the elements of the complex exponential sequence follow the rule
zn= an
zn i the nth member of the sequence, with index n an integer, including 0. The parameter a is a complex number, as is z. With z0 =1
the members are: 1, a, a2, a3, a4..... zn = an
The complex geometric partial sum series Sn is formed by consecutive addition of the members of the sequence:
Sn = Σon a m with 0 ≤ m ≤ n ; Sn = 1 + a + a 2....+ an
This simulation calculates 500 elements of the sequence. By drawing the red point in the left chart, a is defined. The blue points are the members of the sequences, as they are of the partial sum series in the right chart.
The case of the real geometric series is observed when a is a point on the real axis.
The complex geometric series converges if the absolute value of a is smaller than 1, a lying inside of the red unit circle. If convergent, the limit is
limn→∞ Sn = limn→∞ Σon am = 1/(1-a)
It is at the center of the small green circle in the right chart of the series.
The marked point at the unit circle is the first term of both sequence and series, the real number 1. The prominent second point in the sequence is a, which can be drawn with the mouse.
When a has an imaginary component and abs(a)<1 sequence and series spiral towards the convergence point (limit). When small imaginary parts are chosen the whole series will be on one Riemann sheet. For larger imaginary parts the spiral can have many revolutions, the points of which cover more than one Riemann sheet. The spiraling is much more pronouncedly visible than with the exponential series: because of the slower convergence many calculated points are distinguishable with the geometric series.
When a approaches the unit circle from the inside complex patterns may be observed as the series converges. When the real part is negative, the spiral splits into several arms (the real one series splits in two). An interesting split is observed at a real part of zero and a close to i. When a crosses the unit circle the series diverges.
For angles of the a vector of 2π/N, with N integer, the spirals have N- fold symmetry. This is best seen when a is close to the unit circle.
The series increases very fast as a approaches the unit circle. The scales are self adjusting and the unit circle may appear just like a point.
For abs(a) > 0 the series diverges. The unit circle becomes invisible and the series diverges in spirals to infinity. For better viewing draw the diagram up to full screen size.
Near the inner rim of the unit circle convergence may be so slow that 500 calculated points are not sufficient to approach the series limit at the center of the green circle.