Chapter 6.3, Problem 130E

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.

130. Integrate the approximation $\sin t\approx t-\frac{t^3}{6}+\frac{t^5}{120}-\frac{t^7}{5040}$ evaluated at $\pi t$ to approximate ${\displaystyle {\int }_0^1\frac{\sin \pi t}{\pi t}dt}$.