The graphic on the left shows the graph of the absolute values of the partial sums

, where the value of

is controlled by the parameter "truncation". The values of the arguments of the function are reflected in the hue of the graph. The graphic on the right (which you can hide to speed up updating) shows the graph of the limit function

. While all the series converge in the interior of the unit disk, the behavior on the boundary depends on the value of

. For

the series never converges on the boundary, for

it converges everywhere except at

, and for

it converges everywhere. In the case

, large values of the truncation parameter give rise to graphs that appear to show discontinuity at the boundary (see the first snapshot). The functions defined by partial sums are, of course, continuous and the apparent discontinuities are due to numerical instability.