Chapter 6.3, Problem 131E

In the following exercises, verify that the given choice of n in the remainder estimate $\left|Rn\right|\le \frac{M}{(n+1)!}{(x-a)}^{n+1}$ where M is the maximum value of $|f^{(n+1)}(z)|$ on the interval between a and the indicated point, yields $\left|R_n\right|\le \frac{1}{1000}$ . Find the value of the Taylor polynomial P_nof f at the indicated point.

131. Integrate the approximation $e^x\approx 1+x+\frac{x^2}{2}+\mathrm{...}+\frac{x^6}{720}$evaluated at −x²to approximate ${\displaystyle {\int }_0^1{e}^{-x^2}dx}$.