Using Euler's formula to evaluate this series

Question

I have the following exercise:

Let $a$ be a complex constant such that $|a|<1$. Use Euler's formula to prove that $$1 + a\cos \theta + a^2 \cos 2\theta + a^3 \cos 3 \theta + ... = \frac{1-a\cos \theta}{1-2a\cos \theta + a^2}$$

I initially tried to define: $$z := ae^{i\theta}$$ And so the series would become: $$\operatorname{Re}(z^0) +\operatorname{Re}(z^1) + \operatorname{Re}(z^2) + \operatorname{Re}(z^3) + ... $$ And then i tried to search for a closed formula for $\operatorname{Re}(z^n)$, but the ones i found seemed to be pretty dirty and messy(i would be doing a infinite sum of partial sums) . So i concluded that the approach of a series of real parts wouldn't be the best approach to the problem.

Any ideas?

Thanks in advance!

Answer 1

Hint: Consider the series $\sum z^n$, find its sum, and then take the real part.