$$ f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n \textcolor{#C0C0C0}{=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2}(x-x_0)^2\cdots} $$
1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}\cdots} $$
$$ \sin x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)!} \textcolor{#C0C0C0}{ =x-\frac{x^3}{6}+\frac{x^5}{120}\cdots} $$
1-\frac{x^2}{2}+\frac{x^4}{24}\cdots} $$
$$ \frac{1}{1-x}=\sum_{n=0}^\infty x^n \textcolor{#C0C0C0}{=1+x+x^2+x^3\cdots} $$
$$ \frac{1}{1+x}=\sum_{n=0}^\infty(-1)^n x^n \textcolor{#C0C0C0}{=1-x+x^2-x^3\cdots} $$
$$ \ln(1+x)=\sum_{n=0}^\infty (-1)^n\frac{x^{n+1}}{n+1} \textcolor{#C0C0C0}{=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots} $$
$$ \frac{1}{(1+x)^2} = \sum_{n=0}^{\infty} (-1)^n (n+1) x^n $$
$$ \frac{1}{(1-x)^2} = \sum_{n=0}^{\infty} (n+1) x^n $$