A-14 Taylor Series

The Taylor series is a mathematical formula to approximate a complex equation as a polynomial. Formally speaking, the Taylor series of a function is an infinite sum of the evaluation of the function’s derivatives at a single point. Given function $f(X)$, its Taylor series evaluated at $X=a$ is expressed as follows:

$f(a)+{\displaystyle \frac{f^{\prime }(a)}{1!}}(X-a)+{\displaystyle \frac{f^{″}(a)}{2!}}{(X-a)}^2+{\displaystyle \frac{f^{‴}(a)}{3!}}{(X-a)}^3+\cdots =\underset{d=0}{\overset{\infty }{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{\displaystyle \frac{f^{(d)}(a)}{d!}}{(X-a)}^d$

For a target function, the Taylor series can aggregate a finite number of terms, $D$, instead of $\infty$ terms. Such a $D$-degree polynomial is also called the $D$-th Taylor polynomial approximating $f(X)$. The higher the total number of degrees $D$ is, the more accurate the approximation of $f(X)$ becomes. The accuracy of the approximation is higher for those coordinates nearby $X=a$, and lower for those coordinates away from $X=a$. To increase the accuracy for further-away coordinates, we need to increase $D$.