Expand a complex function into a series

Question

I need to expand this function $\frac{1}{2i+z}$ into a series, z is a complex number. I initially tried substituting $z = x+iy$ into z, combine imaginary terms, then use binomial theorem. However, the solution gives the series in terms of $z$. I am wondering how one would go about this problem to get a series in terms of $z$?

Answer 1

Hint：

Write $\frac{1}{2i+z} = \frac{1}{2i(1+\frac{z}{2i})} = \frac{1}{2i}\times(1 - \frac{z}{2i} + (\frac{z}{2i})^2 - (\frac{z}{2i})^3 + ...)$

As

$$\frac{1}{1-z} = 1+z+z^2+z^3 + ... = \sum_{i=0}^{\infty}{z^i}$$