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Excuse me, a problem of higher mathematics homework
Solution: Because ρ=n→+∞lim[a? n+ 1? /a? n? ]=n→+∞lim[ 1/(2n+2))! ]/[ 1/(2n)! ]

=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 (-∞, +∞).