- #1

RJLiberator

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## Homework Statement

Let 0 < r < 1. Show that

from n=1 to n=∞ of Σ(r^ncos(n*theta)) = (rcos(theta)-r^2)/(1-2rcos(theta)+r^2)

Hint. This is an example of the statement that sometimes the fastest path to a “real” fact is via complex numbers. Let z = reiθ. Then, since r = |z|, and 0 < r < 1, the series n=1 to n=∞ Σz^n converges to 1/(1−z).

## Homework Equations

z = x+iy

Not really sure what else yet...

## The Attempt at a Solution

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So here, I understand the first step is to look at the sum from n=0 to n=∞ of z^n and that we know converges to 1/(1-z) when |z|<1.

**My question**is how do I split this up into real and imaginary forms to solve this problem.

Do I split it up in the summation such that from n=0 to n=∞ Σ (x+iy)^n = the sum from n=0 to n=∞ Σx^n + the sum from n=1 to n=∞ Σ(iy)^n and then find the solution that way?