• \({e^x} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + ... = \sum\limits_{n = 0}^\infty {\frac{x^n}{n!}}, x \in \mathbb {R}\)

• \(\sin x = x - \frac{{{x^3}}}{{3!}} + \frac{{{x^5}}}{{5!}} - ... = \sum\limits_{n = 0}^\infty {\frac{{{{\left( { - 1} \right)}^n}}}{{\left( {2n + 1} \right)!}}} {x^{2n + 1}}, x \in \mathbb {R}\)

• \(\cos x = 1 - \frac{{{x^2}}}{{2!}} + \frac{{{x^4}}}{{4!}} - ... = \sum\limits_{n = 0}^\infty {\frac{{{{\left( { - 1} \right)}^n}}}{{\left( {2n} \right)!}}} {x^{2n}}, x \in \mathbb {R}\)