Start with the geometric series: ∞∑n=0zn=1+z+z2+z3+⋯.
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.
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 z≠1, 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/(1−z). 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/(1−z) 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/(1−z) 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 1−1+1−1+⋯.
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