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## Main Question or Discussion Point

Hi guys, I'm having trouble conceptualizing Taylor's Theorem. I understand how to compute a taylor series, as well as figure out whether it converges or not. I also realize that since it is an approximation that there is a remainder that is the difference between it and the original function. I understand all that. My main issue of concern is with the algebraic notation of the remainder. In my book, which is written very poorly, it gives the remainder in two dfferent forms:

"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) + ... + R

where

R

Forgive me for not knowing the coding and such. So, the equation(s) might look a bit sloppy.

The book goes on to say:

"One useful consequence of this theorem is that |R

This part does not make sense to me. Again, sorry for making the formulas appear sloppy. I do not quite understand what that last inequality means. And, in addition, I do not know what max |f

"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) + ... + R

_{n}(x)where

R

_{n}(x) = (f^{n+1}z)/(n+1)! (x-c)^{n+1}."Forgive me for not knowing the coding and such. So, the equation(s) might look a bit sloppy.

The book goes on to say:

"One useful consequence of this theorem is that |R

_{n}x| [tex]\leq[/tex] |x-c|^{n+1}/(n+1)! max |f^{n+1}z|"This part does not make sense to me. Again, sorry for making the formulas appear sloppy. I do not quite understand what that last inequality means. And, in addition, I do not know what max |f

^{n+1}z| refers to. Thanks for any help, and for even reading this.