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先解这个：(n+2)/[n!+(n+1)!+(n+2)!]
=(n+2)/[n!*(1+(n+1)+(n+1)(n+2))]
=(n+2)/[n!*(n+2)^2]
=1/[n!(n+2)]
=(n+1)/[n!(n+1)(n+2)]
=(n+1)/(n+2)!
=(n+2-1)/(n+2)!
=1/(n+1)!-1/(n+2)!
∴原式：3/(1!+2!+3!)+4/(2!+3!+4!)+...+(n+2)/[n!+(n+1)!+(n+2)!]
=(1/2!-1/3!)+(1/3!-1/4!)+...+(1/(n+1)!-1/(n+2)!)
=1/2-1/(n+2)!
=(n+2)/[n!*(1+(n+1)+(n+1)(n+2))]
=(n+2)/[n!*(n+2)^2]
=1/[n!(n+2)]
=(n+1)/[n!(n+1)(n+2)]
=(n+1)/(n+2)!
=(n+2-1)/(n+2)!
=1/(n+1)!-1/(n+2)!
∴原式：3/(1!+2!+3!)+4/(2!+3!+4!)+...+(n+2)/[n!+(n+1)!+(n+2)!]
=(1/2!-1/3!)+(1/3!-1/4!)+...+(1/(n+1)!-1/(n+2)!)
=1/2-1/(n+2)!
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