Chapter 9.8, Problem 57E

Using Power Series Let $f(x)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{(-1)}^n{x}^{2n+1}}{(2n+1)!}\text{ and}}g(x)={\displaystyle \underset{n-0}{\overset{\infty }{\sum }}\frac{{(-1)}^n{x}^{2n}}{(2n)!}.}$

(a) Find the intervals of convergence of f and g.

(b) Show that $f\text{'}(x)=g(x)\text{ and }g\text{'}(x)=\text{ –}f(x).$

(c) Identify the functions f and g.