# Understanding Taylor series error function and Lagrange error bound

I am in high school and find math interesting so lately I have been trying to learn as much about it as I can. I recently began studying Taylor Series as it pertains to the research areas that I am currently following and have two questions. I am trying to conceptualize Lagrange's error bound and the remainder function.

$$f(x)=f(c)+f'(c)(x-c)+...+\frac{f^{(n)}(c)}{n!}(x-c)^{n}+R_n(x)$$

where $$R_n(x)$$ is the remainder given by:

$$\frac{f^{(n+1)}(z)}{(n+1)!}(x-c)^{n+1}$$.

From what I have read, I understand that the remainder gives the difference between a given Taylor polynomial and its original function, $$f(x)$$. I have also read about La Grange's error bound which differs from the remainder.

My first question is, if Lagrange's error bound gives the maximum "remainder" for a given $$x$$, how does it differ from the remainder function $$R_n(x)$$? Are they equivalent? If a function can be found from its Taylor polynomial and Remainder term or any combination vice-versa, what is the purpose of Lagrange's error bound?

My second question is, how is the error term function, $$R_n(x)$$ derived? I understand that it can be found from the difference between a function and its Taylor polynomial but I am not sure where it originates from in general form.

Any help with conceptualizing this is greatly appreciated.

$$R_n(x)$$ is defined to be the difference between the function and its $$n$$th order Taylor polynomial (not series).
It is an important consequence of the Mean Value Theorem that there exists $$z$$ between $$x$$ and $$c$$ such that the remainder can be written as $$R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!} (x-c)^{n+1}$$. However in general we do not know more about the exact value of $$z$$ (or $$f^{(n+1)}(z)$$) beyond the fact that it lies between $$x$$ and $$c$$.
Since we often cannot compute the remainder explicitly, we would like to bound it. For example, it would be nice to say $$|R_n(x)| \le 0.01$$; this would tell us that the value of the $$n$$th order Taylor polynomial at $$x$$ is within $$0.01$$ of the actual function value at $$x$$. Error bounds use assumptions on $$f$$ to bound $$|R_n(x)|$$.