If I want to compute the radius of convergence of a given $\sum_\limits{n=1}^{\infty}c_n z^n\:\:z\in\mathbb{C}$ series I have the formula $\lim_{n\to\infty}\frac{c_{n+1}}{c_n}=\frac{1}{R}$ where $R$ stands for the radius of convergence. On Wikipedia I found the following derivation $|z-a|<\frac{1}{\lim_{n\to\infty}\frac{c_{n+1}}{c_n}}=\lim_{n\to\infty}\frac{c_{n}}{c_{n+1}}=R$.
Questions:
The inverse of the limit is not necessarily the inverse of expression under the limit. What does back up this step $\frac{1}{\lim_{n\to\infty}\frac{c_{n+1}}{c_n}}=\lim_{n\to\infty}\frac{c_{n}}{c_{n+1}}$? What is being used? How do I derive that formula?
Thanks in advance!