Complex Power Series/Examples/2n Factorial over n Factorial Squared

Example of Complex Power Series
Let $\sequence {a_n}$ be the sequence defined as:
 * $a_n = \dfrac {\paren {2 n}!} {\paren {n!}^2} z^n$

The complex power series:


 * $S = \displaystyle \sum_{n \mathop \ge 0} a_n z^n$

has a radius of convergence of $\dfrac 1 4$.

Proof
Let $R$ denote the radius of convergence of $S$.

Thus: