Chapter 2.7, Problem 17P

Verify the series in (7.3) by computer. Also show that it can be written in the form ${\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{(-1)}^n{z}^{2n}{\displaystyle \underset{k=0}{\overset{n}{\sum }}\frac{1}{(2k+1)!}}}$

Use this form to show by ratio test that the series converges in the disk |z| < 1.