Формулы Тейлора для основных элементарных функций

$$e^{x} = 1 + x + \dfrac{x^{2}}{2} + \dfrac{x^{3}}{3!} + \dots + \dfrac{x^{n}}{n!} + o(x^{n}) = \sum_{k=0}^{n}\dfrac{x^{k}}{k!} + o(x^{n})$$

$$\sin x = x - \dfrac{x^{3}}{3!} + \dfrac{x^{5}}{5!} - \dfrac{x^{7}}{7!} + \dots + o(x^{2n+2}) = \sum_{k=0}^{n}\dfrac{x^{2k+1}}{(2k+1)!}(-1)^{k} + o(x^{2n+2})$$

$$\cos x = 1 - \dfrac{x^{2}}{2} + \dfrac{x^{4}}{4!} - \dfrac{x^{6}}{6!} + \dots + o(x^{2n+1}) = \sum_{k=0}^{n}\dfrac{x^{2k}}{(2k)!}(-1)^{k} + o(x^{2n+1})$$

$$\ln(x+1) = x - \dfrac{x^{2}}{2} + \dfrac{x^{3}}{3} - \dfrac{x^{4}}{4} + \dots + o(x^{n}) = \sum_{k=1}^{n}\dfrac{x^{k}}{k}(-1)^{k+1} + o(x^{n})$$

$$\begin{gather} (1+x)^{\alpha} = 1 + \alpha x + \dfrac{\alpha(\alpha -1)}{2!}x^{2} + \dfrac{\alpha(\alpha-1)(\alpha-2)}{3!}x^{3} + \dots = \sum_{k=0}^{n} {\alpha\choose k} x^{k} + o(x^{n}) \\ {\alpha\choose k} = \dfrac{\alpha(\alpha-1)(\alpha-2)\dots(\alpha-k+1)}{k!} \end{gather}$$ $\begin{pmatrix}\alpha\\k\end{pmatrix}$ - это как $C^{k}_{n}$, но не только для ${} n \in \mathbb{N} {}$

$$\dfrac{1}{1-x} = 1 + x + x^{2} + \dots + o(x^{n}) = \sum_{k=0}^{n}x^{k} + o(x^{n})$$