Remainder term in Taylor's theorem

Let $f$ be a function for which the Taylor polynomial $T_k f(x;a)$ is defined. The remainder term of order $k$ (about $a$) is the function

Taylor’s theorem gives hypotheses under which $R_k(x;a)$ can be bounded or represented explicitly (e.g., Lagrange form or integral form), making the approximation $f(x)\approx T_k f(x;a)$ quantitatively precise.

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 $a=0$, $R_3(x;0)=\sin x-\left(x-\frac{x^3}{6}\right)$.