[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that [math]\sum_{n=0}^\infty\frac{1}{n!}=e[/math]. Also, prove that [math]\left(1+\frac{x}{n}\right)^n\to e^x[/math], and that [math]\sum_{n=0}^\infty\frac{x^n}{n!}=e^x[/math], for [math]x=-1[/math], then for [math]x\in\mathbb Z[/math], then for [math]x\in\mathbb R[/math].