# How are the Taylor Series derived?

I know the Taylor Series are infinite sums that represent some functions like $\sin(x)$. But it has always made me wonder how they were derived? How is something like $$\sin(x)=\sum\limits_{n=0}^\infty \dfrac{x^{2n+1}}{(2n+1)!}\cdot(-1)^n = x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}\pm\dots$$ derived, and how are they used? Thanks in advance for your answer.

$$\fermi\pars{x} = \fermi\pars{0} + \int_{0}^{x} \fermi’\pars{t}\,\dd t \,\,\,\stackrel{t\ \mapsto\ x – t}{=}\,\,\, \fermi\pars{x} = \fermi\pars{0} + \int_{0}^{x}\fermi’\pars{x – t}\,\dd t$$
\begin{align} \color{#00f}{\fermi\pars{x}}&= \fermi\pars{0} + \fermi’\pars{0}x + \int_{0}^{x}t\fermi”\pars{x – t}\,\dd t \\[5mm] & = \fermi\pars{0} + \fermi’\pars{0}x + \half\,\fermi”\pars{0}x^{2} +\half\int_{0}^{x}t^{2}\fermi”’\pars{x – t}\,\dd t \\[8mm]& = \cdots = \color{#00f}{\fermi\pars{0} + \fermi’\pars{0}x + \half\,\fermi”\pars{0}x^{2} + \cdots + {\fermi^{{\rm\pars{n}}}\pars{0} \over n!}\,x^{n}} \\[2mm] & + \color{#f00}{{1 \over n!}\int_{0}^{x}t^{n} \fermi^{\rm\pars{n + 1}}\pars{x – t}\,\dd t} \end{align}