Taylor polynomial errors

As Taylor polynomials are approximations of actual functions, they carry errors that we should take note of.

Taylor's remainder theorem

For some Taylor polynomial centred at $x=a$, and the function has $n+1$ continuous derivatives on an interval around $a$, then we can write:

$$f(x)=P_n(x)+R_n(x)$$

where $R_n(x)$ is the remainder (or in other words the deviation). We say it is equal to:

$$R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a{)}^{n+1}$$