Remainder term in Taylor's theorem

Suppose $f$ has derivatives through order $k$ at $a$ . Its Taylor polynomial of degree at most $k$ about $a$ is

T_kf(x;a)=\sum_{j=0}^k\frac{f^{(j)}(a)}{j!}(x-a)^j.

The remainder term of order $k$ is

R_k(x;a)=f(x)-T_kf(x;a)

at every $x$ where $f(x)$ is defined.

Taylor's theorem supplies additional hypotheses under which $R_k(x;a)$ has a Lagrange or integral representation, and hence can be estimated.

Examples

If $f$ is a polynomial of degree $\le k$ , then $R_k(x;a)\equiv 0$ for all $x$ .
For $f(x)=e^x$ about $a=0$ , $R_1(x;0)=e^x-(1+x)$ .
For $f(x)=\sin x$ about $a=0$ , $R_3(x;0)=\sin x-\left(x-\frac{x^3}{6}\right)$ .