阿Q吧 > 化简：3⼀(1!+2!+3!)+4⼀(2!+3!+4!)+....+（n+2)⼀[n!+(n+1)!+(n+2)!]

化简：3⼀(1!+2!+3!)+4⼀(2!+3!+4!)+....+（n+2)⼀[n!+(n+1)!+(n+2)!]

要详细的化简过程，非常感谢……

2024-11-29 00:41:31

推荐回答（1个）

回答1：

先看一般项。
(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)!
所以
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)!
即原式=1/2-1/(n+2)!