Taylor series

(Monday, November 1, 2021)

If we let $n\to\infty$ in the $n$ th order Taylor polynomial of $f(x)$ about $x=0$ , we arrive at the Taylor series

\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^{n}.

Similarly, there is a Taylor series about $x=a$ :

\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}.

Note that when $n=0$ and $x=a$ , the above is

\frac{f(0)\cdot 0^{0}}{0!}+\frac{f^{\prime}(0)\cdot 0^{1}}{1!}++\frac{f^{% \prime\prime}(0)\cdot 0^{b}}{2!}+\cdots

by convention, we define $0^{0}=1$ in this context.

Observe that Taylor polynomials are the partial sums of Taylor series.