Complex series radius convergence

Question

How to find the values for which $z$ converges, $z\in\mathbb{C}$, in the serie $$\sum_{n=1}^{\infty}\frac{1}{(1+|z|^{2})^{n}}$$ I know I have to use the convergence radius expression, but what I suppose to do here? I mean, as $$R=\lim_{n\to\infty}\left|\frac{a_{n}}{a_{n+1}}\right|$$ what is $a_{n}$ in this case?

Answer 1

Hint:

For which values of $y$ does

$$ \sum_{n=1}^{\infty} y^n $$

converge?

Now let

$$ y = \frac{1}{1+|z|^2}. $$

Answer 2

Hint This is not really a question about complex numbers. Think geometric series.

For the question as it stands now, you do not have to use the convergence radius expression. But to answer your question about $a_n$, if you will use the Ratio Test then you should use $a_n=\frac{1}{(1+|z|^2)^n}$. Note that our series is not a power series in the usual sense. So formulas you may remember about radius of convergence need not apply.