=n→+∞lim[(2n)! /(2n+2)! ]=n→+∞lim[(2n)! /(2n)! (2n+ 1)(2n+2)]
= n →+∞lim[ 1/(2n+ 1)(2n+2)]= 0
So the convergence radius R= 1/ρ=∞, and the convergence interval is (-∞, +∞).