Given that:

$cosx={\displaystyle \underset{n=0}{\overset{}{}}}\frac{(-1{)}^n{x}^{2n}}{(2n)!},$ for $|x|<mi></mi>$

a$.$ Find the power series representation for $f(x)=cos(x^3)$

b$.$ Find the series representation for ${\displaystyle \underset{0}{\overset{0\mathrm{.}4}{}}}cos(x^3)dx$

c$.$ How many terms of the series for ${\displaystyle \underset{0}{\overset{0\mathrm{.}4}{}}}cos(x^3)dx($The series you found in part b$)$ are needed to estimate ${\displaystyle \underset{0}{\overset{0\mathrm{.}4}{}}}cos(x^3)dx$ with an $|||$ error Hint: Use the Alternating Series Test Estimation theorem.$)$