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Friday, May 28, 2021

a geometric interpretation for the sum of a geometric series

Start with the geometric series: n=0zn=1+z+z2+z3+.

If z is not a real number, the partial sums of this series form a spiral pattern, which appears to be self-similar. Below is the pattern for z=(1+i)/2. (Click for interactive graph.)

A similarity of the plane can be expressed as a (complex) affine function f(w)=aw+b. To determine a and b for our spiral, we note that f should satisfy f(0)=1andf(1)=1+z.

The first equation implies b=1. From the second equation, we then have a+1=1+z, so a=z. Our function is therefore f(w)=zw+1, with z treated as a constant.

Now we check whether this function f(w) does in fact show that the spiral is self-similar. Indeed, f(1+z)=z(1+z)+1=1+z+z2, and more generally, the sequence 1, f(1), f(f(1)), f(f(f(1))), … coincides with the sequence of partial sums of the geometric series.

If z1, then the function f(w) has a fixed point. Solve f(w)=w, or zw+1=w, to find that the fixed point is w=1/(1z). When |z|<1, this fixed point is the sum of the series. But in general it has a nice geometric connection to the series, even when the series diverges: it is the center of similarity for the sequence of partial sums. For example, when z=i+1/3, the partial sums and center look like this:

In particular, when z is a root of unity, the partial sums of the series lie at the vertices of a regular polygon, and 1/(1z) is the center of this polygon. (It is also the Cesàro sum of the series in that case.) Here are the polygons and centers for z=eiπ/4 and z=ei3π/5:

The centers of these polygons all have real part 1/2, which is a special case of the observation that the function 1/(1z) sends the unit circle to the line Rez=1/2.

When z=1, the polygon collapes to a line segment, and of course the center of this segment is 1/2, the usual value assigned to the series 11+11+.

1 comment:

Ufa88kh said...

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