Chapter 8.2, Problem 7E

To determine

To show:

The series $a_0+a_2{x}^2+a_4{x}^4+a_6{x}^6+\mathrm{...}={\displaystyle \sum _{k=0}^{\infty }{a}_{2k}{x}^{2k}}$ has a radius of convergence $\rho =\sqrt{L}$, if

$\underset{n\to \infty }{\lim }\left|\frac{a_{2k}}{a_{2k+2}}\right|=L$, and that $a_{1}x+a_3{x}^3+a_5{x}^5+a_7{x}^7+\mathrm{...}={\displaystyle \sum _{k=0}^{\infty }{a}_{2k+1}{x}^{2k+1}}$ has a radius of convergence $\rho =\sqrt{M}$, if $\underset{n\to \infty }{\lim }\left|\frac{a_{2k+1}}{a_{2k+3}}\right|=M$.