Does this sequence of complex numbers converge? I need to determine if the following recursive sequence of complex numbers converges.
$$z_0 = 0, \quad z_n = \frac{1}{2i + z_{n-1}}, \quad n \geq 1.$$
I started by expanding the sequence:
$$z_1 = \frac{1}{2i}, \quad z_2 = \frac{1}{2i + \frac{1}{2i}}, \quad z_3 = \frac{1}{2i + \frac{1}{2i + \frac{1}{2i}}}, \dots, $$
so this is a continued fraction. What strategies can I use to determine if a sequence of complex numbers converges or not?

It helps to actually compute the first few terms. Using $i^{-1}=-i$, we get that
$$\begin{cases}z_0 = 0 \\ z_1 = -\frac{1}{2}i \\ z_2 = -\frac{2}{3}i \\ z_3 = -\frac{3}{4}i \\ \cdots\end{cases}$$
so it is relatively straightforward to conjecture (and prove via induction, an exercise for OP) that
$$z_n = -\frac{n}{n+1}i$$
