# Origin of Taylor Series

Historically, the Taylor series representations or truncated Taylor series approximations of a function at a point $$x_0$$ was first done by taking Newton's form of an interpolation polynomial for points of the form $$x_0 + n \Delta$$, where $$\Delta$$ is a positive real number and $$n$$ is a natural number, and then taking the limit as $$\Delta$$ goes to $$0$$. Could someone explain in detail how this was done?

Let $$f$$ be a function on an interval of real numbers. Let $$\Delta$$ be a real number and let $$x_0$$ be in the domain of $$f$$ such that $$x_0$$, $$x_0 + \Delta$$, $$\dots$$, $$x_0 + n\Delta$$ is in the domain of $$f$$, where $$n$$ is a positive integer. Newton's form of the interpolation polynomial of the data $$\{ (x_0 + k \Delta, f(x_0 + k\Delta) \}$$ is $$f(x_0) + \frac{f(x_0 + \Delta) - g_0(x_0+\Delta)}{\Delta}(x-x_0) + \cdots + \frac{f(x_0 + n\Delta ) - g_{n-1}(x_0 + n\Delta)}{n!\Delta}(x-x_0)\cdots (x-(n-1)\Delta),$$ where $$g_k$$ is the interpolation polynomial for the first $$k+1$$ points. Taking the limit as $$\Delta$$ tends towards $$0$$ gives $$f(x_0) + f'(x_0)\cdot x + \cdots + \frac{1}{n!}\cdot (\lim_{\Delta \to 0} \frac{f(x_0 + n\Delta) -g_{n-1}(x_0 + n\Delta)}{\Delta^n})\cdot x^n,$$ as long as $$f$$ is sufficently smooth and the limit $$\lim_{\Delta \to 0} \frac{f(x_0 + n\Delta) -g_{n-1}(x_0 + n\Delta)}{\Delta^n}$$ exits. But why does the limit $$\lim_{\Delta \to 0} \frac{f(x_0 + n\Delta) -g_{n-1}(x_0 + n\Delta)}{\Delta^n}$$ equal $$f^{(n)}(x_0)$$?

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\begin{align} \mrm{f}\pars{x} & = \mrm{f}\pars{0} + \int_{0}^{x}\mrm{f}'\pars{t}\dd t \,\,\,\stackrel{t\ \mapsto\ x - t}{=}\,\,\, \mrm{f}\pars{0} + \int_{0}^{x}\mrm{f}'\pars{x - t}\dd t \\[5mm] & \stackrel{\mrm{IBP}}{=}\,\,\, \mrm{f}\pars{0} + \mrm{f}'\pars{0}x + \int_{0}^{x}\mrm{f}''\pars{x - t}t\,\dd t \\[5mm] & \stackrel{\mrm{IBP}}{=}\,\,\, \mrm{f}\pars{0} + \mrm{f}'\pars{0}x + {1 \over 2}\,\mrm{f}''\pars{0}x^{2} + {1 \over 2}\int_{0}^{x}\mrm{f}'''\pars{x - t}t^{2}\,\dd t \\[1cm] & \stackrel{\mrm{IBP}}{=}\,\,\, \mrm{f}\pars{0} + \mrm{f}'\pars{0}x + {1 \over 2}\,\mrm{f}''\pars{0}x^{2} + {1 \over 6}\,\mrm{f}'''\pars{0}x^{3} \\[2mm] &\ + {1 \over 6}\int_{0}^{x}\mrm{f}^{\pars{\texttt{IV}}}\pars{x - t}t^{3}\,\dd t \\[1cm] & = \cdots = \bbx{\sum_{k = 0}^{n}\mrm{f}^{\mrm{\pars{k}}}\pars{0}\,{x^{k} \over k!} + {1 \over n!}\ \underbrace{\int_{0}^{x} \mrm{f}^{\mrm{\pars{n + 1}}}\pars{x - t}t^{n}\dd t} _{\ds{\int_{0}^{x} \mrm{f}^{\mrm{\pars{n + 1}}}\pars{t}\pars{x - t}^{n}\,\dd t}}} \end{align}