Math $2205-$ Written Homework $10$

Due $12/6$ at $11$:$59$ pm

a$)$ Use the known Taylor series for cosine to find the Taylor series centered at $0$ of $f(x)=cos(x^2).$ Express your answer in summation notation and list the first three non$-$zero terms of the series. Simplify your answer.

b$)$ Use the Taylor series you found in part a$)$ to evaluate the following limit$.$

$\underset{x0}{lim}\frac{2cos(x^2)-2-x^4}{x^8}$

c$)$ Use the Taylor series you found in part a$)$ to represent ${\displaystyle \underset{0}{\overset{1}{}}}cos(x^2)dx$ as an infinite series in summation notation.

d$)$ Use the series found in part c$)$ to approximate ${\displaystyle \underset{0}{\overset{1}{}}}cos(x^2)dx$ with an error less than ${10}^{-3}.$ Be sure to justify that your approximation satisfies the error bound.