1$e^x$
$\sum_{k=0}^n\frac{x^k}{k!} + o(x^n) = 1 + x + \frac{x^2}{2} + ... + \frac{x^n}{n!} + o\left(x^n\right)$
2$(1+x)^\alpha$
$\sum_{k=0}^n \binom{\alpha}{k}x^k + o(x^n) = 1 + \alpha x + \frac{\alpha(\alpha - 1)}{2}x^2 + ... + \frac{\alpha(\alpha - 1)...(\alpha - x + 1)}{n!}x^n + o(x^n)$
3$\frac{1}{1-x}$
$\sum_{k=0}^n x^k + o(x^n) = 1 + x + x^2 + ... + x^n + o(x^n)$
4$\frac{1}{1+x}$
$\sum_{k=0}^n (-1)^k x^k + o(x^n) = 1 - x + x^2 - ... + (-1)^n x^n + o(x^n)$
5$\ln(1 + x)$
$\sum_{k=1}^n(-1)^k \frac{x^k}{k} + o(x^n) = x - \frac{x^2}{2} + \frac{x^3}{3} + ... + (-1)^{n-1}\frac{x^n}{n} + o(x^n)$
6$arctan(x)$
$\sum_{k=0}^n (-1)^k \frac{x^{2k+1}}{2k+1} + o(x^{2n + 2}) = x - \frac{x^3}{3} + ... + (-1)^n \frac{x^{2n+1}}{2n+1} + o(x^{2n + 1})$
7$tan(x)$
$x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + o(x^8)$
8$cos(x)$
$\sum_{k=0}^n (-1)^k \frac{x^{2k}}{(2k)!} + o(x^{2n+1}) = 1 - \frac{x^2}{2} + ... + (-1)^n \frac{x^{2n}}{(2n)!} + o(x^{2n+1})$
9$sin(x)$
$\sum_{k=0}^n (-1)^k \frac{x^{2k+1}}{(2k+1)!} + o(x^{2n+2}) = x - \frac{x^3}{6} + ... + (-1)^n \frac{x^{2n+1}}{(2n+1)!} + o(x^{2n+2})$
10$ch(x)$
$\sum_{k=0}^n \frac{x^{2k}}{(2k)!} + o(x^{2n+1}) = 1 + \frac{x^2}{2} + ... + \frac{x^{2n}}{(2n)!} + o(x^{2n+1})$
11$sh(x)$
$\sum_{k=0}^n \frac{x^{2k+1}}{(2k+1)!} + o(x^{2n+2}) = x + \frac{x^3}{6} + ... + \frac{x^{2n+1}}{(2n+1)!} + o(x^{2n+2})$
12$th(x)$
$x-\frac{x^3}{3}+\frac{2x^5}{15}-\frac{17x^7}{315} + (x^8)$
13$arcsin(x)$
$\sum_{k=0}^n \frac{(2k)!}{2^{2k}(k!)^2} \times \frac{x^{2k+1}}{2k + 1} + o(x^{2n+2}) = x + ... + \frac{1 \times 3 \times ... \times (2n-1)}{2 \times 4 \times ... \times (2n)} \frac{x^{2n+1}}{2n+1} + o(x^{2n+2})$
14$arccos(x)$
$\frac{\pi}{2} - arcsin(x)$