Convergence of a "rotating movement" in the Complex Plane During a class, my teacher gave us the following example:

Consider the geometric progression with start value $i$ and common ratio $\frac{i}{2}$. Now get the geometric series of this progression
$$\sum_{n \geq 1} i\left(\frac{i}{2}\right)^n$$
and see what happens in the complex plane: at every term of the sum, the direction of the movement will rotate 90º and the series will converge into a point.

Now what if I want to calculate the point a simple series like that converges, but with a small difference: the movement rotates 39º (or anything more difficult than the "standart angles"). What should I do? I mean, we can represent sines and cosines of an angle using
$$\cos (\theta) = \frac{(e^{i\theta} + e^{-i\theta})}{2} $$$$\sin (\theta) = \frac{(e^{i\theta} - e^{-i\theta})}{2i}$$
but can I represent this movement (rotating 39º) only by a single complex common ratio in a geometric progression?
 A: Well, multiplication by $i=\cos{\frac\pi2}+i\sin{\frac\pi2}=e^{i\pi/2}$ gives you a ninety-degree rotation, so what would you guess gives a rotation by an angle of $\theta$?
A: Notice that $i = \exp\left(i\frac{\pi}{2}\right)$. The fact that terms are rotated $90^{\circ}$ (that is, $\frac{\pi}{2}$) is no coincidence. The $\frac{1}{2}$ that is multiplying $i$ in the ratio only scales the result afterwards.
In general, multiplication by $\exp(i\theta)$ results in a rotation of $\theta$ in the complex plane; you can check this using Euler's formula $\exp(i\theta) = \cos(\theta) + i\cdot\sin(\theta)$. So the ratio of your geometric series will look something like
$$r\cdot\exp\left(i\frac{39\pi}{180}\right)$$
where $0\leq r < 1$ is a real number (the modulus of your ratio). If your question is whether such a ratio admits a 'pretty' expression of the form $a+b\cdot i$, this hinges on your ability to express $\cos(39^{\circ})$ and $\sin(39^{\circ})$ prettily. Any integer degree angle admits fractional expressions in terms of radicals, although they may become a bit too involved to be pretty. For instance,$$\sin(18^{\circ})=\frac{\sqrt{10+2\sqrt{5}}}{4}$$
