# 10.1 | Taylor Polynomials

The Taylor polynomial of degree $$n$$ centered at $$x=a$$ or the Taylor polynomial about $$x =a$$ is given by
$$P_n(x) = f(a) + f'(a) (x-a) + \frac{f^{\prime \prime}(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$$
and $$f(x) \approx P_n(x)$$ near $$x = a$$.

Find the first, second, third, fourth, fifth, and sixth degree Maclaurin polynomial for
$$f(x) = e^x.$$

The $$n^{th}$$-degree Maclaurin polynomial is given by:
$$M_n(x) = \sum_{i = 0}^n \frac{f^{(i)}(0)}{i!}x^i$$

$$\displaystyle i$$ $$\displaystyle f^{(i)}(x)$$ $$\displaystyle \frac{f^{(i)}(0)}{i!}x^i$$
0 $$e^x$$ $$1$$
1 $$e^x$$ $$x$$
2 $$e^x$$ $$\frac{1}{2}x^2$$
3 $$e^x$$ $$\frac{1}{3!}x^3$$
4 $$e^x$$ $$\frac{1}{4!}x^4$$
5 $$e^x$$ $$\frac{1}{5!}x^5$$
6 $$e^x$$ $$\frac{1}{6!}x^6$$

Using the table on the left, we obtain:
\begin{aligned} M_1(x) & = 1 + x \\ M_2(x) & = 1 + x + \frac{x^2}{2} \\ M_3(x) & = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} \\ M_4(x) & = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!}\\ M_5(x) & = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!}\\ M_6(x) & = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!}\\ \end{aligned}