1) Is the remainder you get $R_{n}(x)$ = $f(x)$ - $P_{n}$(x) or is $R_{n}(x)$ just an approximation for the error?
2) Is $R_{n}(x)$ =$\frac{f^{n+1}(x^*)(x-c)^{n+1}}{(n+1)!}$ the rest of the Taylor series from the $(n+1)^{th}$ term with the new variable $x^*$? If so how come it's just one term if it contains the remainder of the Taylor series?
3) Why do you have to pick $x^*$ so that $R_{n}(x)$ will have the biggest error?
Thanks for any replies.
(1) The remainder is by definition the error.
(2) $x^*$ is an unknown point. Isn't more mysterious than the unknown point in the Lagrange's mean value theorem. If what worries you is the size ("how come it's just one term if it contains the remainder of the Taylor series"), don't worry. Being the sum of many things $\ne$ being big.