(related to Proposition: Complex Exponential Function)
From the ratio test for absolutely convergent complex series it follows that
\[\left|\frac{\frac{z^{n+1}}{(n+1) ! }}{\frac{z^n}{n ! }}\right|=\frac{|z|}{n+1}\le\frac 12.\]
Therefore, the complex exponential series
\[\sum_{n=0}^\infty\frac{z^n}{n!}\]
is an absolutely convergent complex series for every complex number \(z\in\mathbb C\), where \(n!\) denotes the factorial of the index \(n\).