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.
I have read that:
where $R_n(x)$ is the remainder given by:
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.