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