When is the binomial expansion with negative exponent valid for polynomials with complex numbers

Question

So for a normal binomial expansion with negative exponent, say

$$f(x) = (1 - 2x)^{-1},$$

we know it is valid for

$$-1/2 < x < 1/2.$$

But what is the case for complex numbers?

I was able to see how the expansion would work in the below link: Binomial Expansion negative power with complex numbers

But I was not able to understand for which values it would be valid.

So say for example,

$$f(x) = (1 - (1+3i)x)^{-1}$$

How can I determine for which values of $x$ it will be valid?

Answer 1

The expansion of a :

$$\frac{1}{1-z}$$

where $z$ is a complex variable is finite when the modulus of $z$ is less than 1.

In your example you have to solve the inequality

$$|(1+3i)x|<1$$

to find the domain. Here it is valid for :

$$|x|<\frac{1}{\sqrt{10}}. $$