quick way to the closed form

A quick (complex) derivation
of the closed form
Mathematical Details (and comments) for
an article by Mandelbrot and Frame:
Self-Contacting Fractal Trees.
by Don West, Department of Mathematics, SUNY Plattsburgh

A colleague, Sam Northshield, suggested looking at the cosine sum

as the real part of a geometric series of exponentials, using

Now we "rationalize the denominator" (actually real-ize):

Again using

we see that

so the denominator is now real, and we have

as we observed earlier.

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