Chapter 12.12, Problem 1P

Show by the ratio test that the infinite series (12.9) for $J_p(x)$ converges for all $x$.

(12.9) $J_p(x)=\frac{1}{\Gamma (1)\Gamma (1+p)}{\left(\frac{x}{2}\right)}^p-\frac{1}{\Gamma (2)\Gamma (2+p)}{\left(\frac{x}{2}\right)}^{2+p}$ $\begin{array}{ll}\hfill & +\frac{1}{\Gamma (3)\Gamma (3+p)}{\left(\frac{x}{2}\right)}^{4+p}-\frac{1}{\Gamma (4)\Gamma (4+p)}{\left(\frac{x}{2}\right)}^{6+p}+\cdots \hfill \end{array}$ $={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{(-1)}^n}{\Gamma (n+1)\Gamma (n+1+p)}}{\left(\frac{x}{2}\right)}^{2n+p}$.