Complex Geometric Series JavaScript Simulation Applet HTML5
The geometric series is written as:
,
where a is the first term of the sequence, and r is a constant real number called the common ratio.
The series converges to a finite value if |r|<1, and the limit that the series converges to in that case is given by 1/(1-a).
This definition still holds for when the common ratio is a complex number z, with the only difference being that instead of having the absolute value of r be less than 1 for the series to converge, the modulus of z has to be less than 1. This is shown in the simulation as the unit circle. As long as the common ratio (denoted a in the simulation) remains inside the unit circle, the series will converge to a finite value. Otherwise, it diverges.
There are two graphs in the simulation (link here). The leftmost graph shows the terms of the series, the colour turning from red to cyan as the terms progress. The rightmost graph shows the partial sums up to each term in the series, and the green square denotes the limit of the series. The colouring of the points in the rightmost graph is the same as the leftmost graph, to make the mapping clearer.
The first term in the series is fixed at z = 1.
There is also an exponential function, where the terms in the series also converge, but they do so at a much slower rate than the geometric series.
,where a is the first term of the sequence, and r is a constant real number called the common ratio.
The series converges to a finite value if |r|<1, and the limit that the series converges to in that case is given by 1/(1-a).
This definition still holds for when the common ratio is a complex number z, with the only difference being that instead of having the absolute value of r be less than 1 for the series to converge, the modulus of z has to be less than 1. This is shown in the simulation as the unit circle. As long as the common ratio (denoted a in the simulation) remains inside the unit circle, the series will converge to a finite value. Otherwise, it diverges.
There are two graphs in the simulation (link here). The leftmost graph shows the terms of the series, the colour turning from red to cyan as the terms progress. The rightmost graph shows the partial sums up to each term in the series, and the green square denotes the limit of the series. The colouring of the points in the rightmost graph is the same as the leftmost graph, to make the mapping clearer.
The first term in the series is fixed at z = 1.
There is also an exponential function, where the terms in the series also converge, but they do so at a much slower rate than the geometric series.
