# Taylor formula, notation for remainder

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 $$\frac{h^q}{q!}\varphi^{(q)}(x+\theta h)=\varphi(x+h)-\varphi(x)-h\varphi'(x)-\cdots-\frac{h^{q-1}}{(q-1)!}\varphi^{(q-1)}(x)$$

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 $$R_{q-1}(x+h)=\frac{h^q}{q!}\varphi^{(q)}(\xi),\quad x<\xi This supposes $$h>0$$; if $$h<0$$, the condition is $$x+h<\xi. 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 $$h$$.