> [!NOTE] Definition > The series (Euler's number) given by $e=\sum_{n=0}^{\infty} \frac{1}{n!}$ [[Convergent Real Series|converges]]. *Proof*. This sum converges by the [[Comparison Test for Series With Non-Negative Terms|comparison test]], since for all $n \geq 2$: $\frac{1}{n!}=\frac{1}{n\times n-1 \times\dots 2} \leq \frac{1}{\underbrace{ 2\times 2 \times\dots 2 }_{\text{n-1 times}}} = \frac{1}{2^{n-1}}$and the [[Convergent Geometric Series|geometric series]] $\sum \frac{1}{2^{n-1}}$ converges.