sequences and series - Sum of two infinite complex geometric sums

let $$C=\frac{\cos\theta}{2}-\frac{\cos2\theta}{4}+\frac{\cos3\theta}{8}+...$$
$$S=\frac{\sin\theta}{2}-\frac{\sin2\theta}{4}+\frac{\sin3\theta}{8}+...$$
I want to find the sum of the series $C+iS$ and thus find expressions for $C$ and $S$.

so the sum of $C+iS$ is
$$C+iS=(\frac{\cos\theta}{2}-\frac{\cos2\theta}{4}+\frac{\cos3\theta}{8}+...)+i(\frac{\sin\theta}{2}-\frac{\sin2\theta}{4}+\frac{\sin3\theta}{8}+...)$$
$$=\frac{1}{2}(cos\theta+i\sin\theta)-\frac{1}{4}(cos2\theta+i\sin2\theta)+\frac{1}{8}(cos3\theta+i\sin3\theta) + ...$$
$$=\frac{1}{2}(cos\theta+i\sin\theta)-\frac{1}{4}(cos\theta+i\sin\theta)^2+\frac{1}{8}(cos\theta+i\sin\theta)^3 + ...$$

So I know i have to now put his into a geometric series where i can find the first term and common ratio as i will want to sum to infinity, but I'm not sure where to go from here?

So I've continued and got the series to here:
$$\frac{1}{2}[e^{i\theta}-\frac{1}{2}(e^{i\theta})^2+\frac{1}{4}(e^{i\theta})^3...]$$ But I'm still not sure where to continue.

Answer

HINT

We have that

$$C+iS=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{2^k}e^{\left(ik\theta\right)}
=-\sum_{k=1}^\infty\left(-\frac{e^{\left(i\theta\right)}}{2}\right)^k$$

then refer to geometric series which holds also for $r$ complex $|r|<1$.