Taylor polynomial

A Taylor polynomial of order $n$ for a function $f$ at a point $a$ is the polynomial $T_n f(x)=\sum_{j=0}^n \frac{f^{(j)}(a)}{j!}(x-a)^j$, assuming the needed higher derivatives exist at $a$.

Taylor polynomials package derivative data at a point into a single polynomial that locally approximates $f$. The approximation is quantified by Taylor's theorem with remainder .

Examples:

• For $f(x)=e^x$ at $a=0$, $T_n f(x)=\sum_{j=0}^n \frac{x^j}{j!}$.
• For $f(x)=\sin x$ at $a=0$, $T_{2n+1} f(x)=\sum_{j=0}^n (-1)^j \frac{x^{2j+1}}{(2j+1)!}$.