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Hi I am wondering how this Taylor formula came about, because I don't really see a remainder term here. Also not sure where the $\theta$ is coming here from. Is this some different way of notation for a remainder?

[...] among the fundamental functions $\varphi(x)$ (see Section 8, below). If $\beta\leqslant1$, then all the functions $\varphi(x)\in S^\beta$ are already analytic and there are, thereby, no functions of bounded support among them. Indeed, the remainder term of the Taylor formula $$\frac{h^q}{q!}\varphi^{(q)}(x+\theta h)=\varphi(x+h)-\varphi(x)-h\varphi'(x)-\cdots-\frac{h^{q-1}}{(q-1)!}\varphi^{(q-1)}(x)$$

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The remainder for the expansion of the Taylor's formula at order $q-1$ of $f(x+h)$ is the left-hand side. Lagrange's form of the remainder is that $$R_{q-1}(x+h)=\frac{h^q}{q!}\varphi^{(q)}(\xi),\quad x<\xi<x+h$$ This supposes $h>0$; if $h<0$, the condition is $x+h<\xi<x$. Both cases are covered by a single formula: any number between $x$ and $x+h$, can be written as $\xi=x+\theta h,\enspace 0<\theta<1,\:$ independently of the sign of $h$.

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