The following theorem is stated in my book.
If a function f is differentiable through order n + 1 in an interval I containing c, then, for each x in I, there exists z between x and c such that $f(x) = f(c) + f'(c)(x - c) + \frac{f''(c)}{2!}(x - c)^2 + ... + \frac{f^n(c)}{n!}(x - c)^n + R_n(x)$ where $R_n(x) = \frac{f^{n + 1}(z)}{(n + 1)!}(x - c)^{n + 1}$
My question is whether z is a fixed number for each n or if it depends on n itself? For example, is the z in $R_7(x)$ generally the same z in $R_{12}(x)$ or might it vary with n?
Take for instance $$f(x)=x^3+1$$ then for $ x>0 $, $$f(x)=f(0)+xf'(z_1)$$ and $$f(x)=f(0)+xf'(0)+\frac{x^2}{2}f''(z_2)$$
So
$$x^3+1=1+x(3z_1^2)\implies z_1=\frac {x}{\sqrt{3}}$$ and
$$x^3+1=1+\frac{x^2}{2}(6z_2)\implies $$ $$z_2=\frac{x}{3}$$
So, they are not the the same.