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.

Answer

$\newcommand{\+}{^{\dagger}}
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Note that
$$
\fermi\pars{x} = \fermi\pars{0} + \int_{0}^{x}
\fermi’\pars{t}\,\dd t
\,\,\,\stackrel{t\ \mapsto\ x – t}{=}\,\,\,
\fermi\pars{x} = \fermi\pars{0} + \int_{0}^{x}\fermi’\pars{x – t}\,\dd t
$$

Integrating by parts:
\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}

Attribution
Source : Link , Question Author : TrueDefault , Answer Author : Felix Marin

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