Taylor Series

Taylor Series around the point $x_{0} = 0$ for basic elementary functions.

$e^{x} = 1 + x + \frac{x^{2}}{2!} + . . . + \frac{x^{n}}{n!} + o (x^{n}) = \sum_{k = 0}^{n} \frac{x^{k}}{k!} + o (x^{k}); x \to 0$

$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + . . . + \frac{(- 1)^{n} x^{2 n + 1}}{(2 n + 1)!} + o (x^{2 n + 2}) = \sum_{k = 0}^{n} (- 1)^{k} \frac{x^{2 k + 1}}{(2 k + 1)!} + o (x^{2 n + 2}); x \to 0 y$

$\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + . . . + (- 1)^{n} \frac{x^{2 n}}{(2 n)!} + o (x^{2 n + 1}) = \sum_{k = 0}^{n} (- 1)^{k} \frac{x^{2 k}}{(2 k)!} + o (x^{2 n + 1}); x \to 0$

$(1 + x)^{α} = 1 + α x + \frac{α (α - 1)}{2!} x^{2} + . . . + \frac{α (α - 1) . . . (α - (n - 1))}{n!} x^{n} + o (x^{n}); x \to 0$

$\frac{1}{1 + x} = \sum_{k = 0}^{n} (- 1)^{k} x^{k} + o (x^{n}); x \to 0$

$\frac{1}{1 - x} = \sum_{k = 0}^{n} x^{k} + o (x^{n}); x \to 0$

$\log (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} + . . . + \frac{(- 1)^{n - 1} x^{n}}{n} + o (x^{n}) = \sum_{k = 0}^{n} \frac{(- 1)^{k - 1} x^{k}}{k} + o (x^{n}); x \to 0$