Activity $8\mathrm{.}5\mathrm{.}5.$ In this activity we encounter several different alternating series and approximate the value of each using the Alternating Series Estimation Theorem.

a$.$ Use the fact that $sin(x)=x-\frac{1}{3!}{x}^3+\frac{1}{5!}{x}^5-$cdots to estimate $sin(1)$ to within $0\mathrm{.}0001.$ Do so without entering $"sin(1)"$ on a computational device. After you find your estimate, enter $"sin(1)"$ on a computational device and compare the results.

b$.$ Recall our recent work with ${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^2}dx$ in Equation $(8\mathrm{.}5\mathrm{.}4),$ which states

${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^2}dx=1-\frac{1}{3}+\frac{1}{5*2!}-\frac{1}{7*3!}$

Use this series representation to estimate ${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^2}dx$ to within $0\mathrm{.}0001.$ Then, compare what a computational device reports when you use it to estimate the definite integral.

c$.$ Find the Taylor series for $cos(x^2)$ and then use the Taylor series and to estimate the value of ${\displaystyle \underset{0}{\overset{1}{}}}cos(x^2)dx$ to within $0\mathrm{.}0001.$ Compare your result toActivity $8\mathrm{.}5\mathrm{.}5.$ In this activity we encounter several different alternating series and approximate the value of each using the Alternating Series Estimation Theorem.

a$.$ Use the fact that $sin(x)=x-\frac{1}{3!}{x}^3+\frac{1}{5!}{x}^5-$cdots to estimate $sin(1)$ to within $0\mathrm{.}0001.$ Do so without entering $"sin(1)"$ on a computational device. After you find your estimate, enter $"sin(1)"$ on a computational device and compare the results.

b$.$ Recall our recent work with ${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^2}dx$ in Equation $(8\mathrm{.}5\mathrm{.}4),$ which states

${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^2}dx=1-\frac{1}{3}+\frac{1}{5*2!}-\frac{1}{7*3!}$

Use this series representation to estimate ${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^2}dx$ to within $0\mathrm{.}0001.$ Then, compare what a computational device reports when you use it to estimate the definite integral.

c$.$ Find the Taylor series for $cos(x^2)$ and then use the Taylor series and to estimate the value of ${\displaystyle \underset{0}{\overset{1}{}}}cos(x^2)dx$ to within $0\mathrm{.}0001.$ Compare your result toActivity $8\mathrm{.}5\mathrm{.}5.$ In this activity we encounter several different alternating series and approximate the value of each using the Alternating Series Estimation Theorem.

a$.$ Use the fact that $sin(x)=x-\frac{1}{3!}{x}^3+\frac{1}{5!}{x}^5-$cdots to estimate $sin(1)$ to within $0\mathrm{.}0001.$ Do so without entering $"sin(1)"$ on a computational device. After you find your estimate, enter $"sin(1)"$ on a computational device and compare the results.

b$.$ Recall our recent work with ${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^2}dx$ in Equation $(8\mathrm{.}5\mathrm{.}4),$ which states

${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^2}dx=1-\frac{1}{3}+\frac{1}{5*2!}-\frac{1}{7*3!}$

Use this series representation to estimate ${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^2}dx$ to within $0\mathrm{.}0001.$ Then, compare what a computational device reports when you use it to estimate the definite integral.

c$.$ Find the Taylor series for $cos(x^2)$ and then use the Taylor series and to estimate the value of ${\displaystyle \underset{0}{\overset{1}{}}}cos(x^2)dx$ to within $0\mathrm{.}0001.$ Compare your result to