Chapter 6.2, Problem 63E

If $f(x)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{x^n}{n!}}$ and $g(x)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{(-1)}^n\frac{x^n}{n!}}$ , find the power series of $\frac{1}{2}\left(f(x)+g(x)\right)$ and $\frac{1}{2}\left(f(x)-g(x)\right)$ .