化简1/2!+2/3!+3/4!……+n-1/n!
来源:学生作业帮 编辑:搜搜做题作业网作业帮 分类:数学作业 时间:2024/07/10 10:16:33
化简1/2!+2/3!+3/4!……+n-1/n!
n大于等于2
n属于N*
n大于等于2
n属于N*
![化简1/2!+2/3!+3/4!……+n-1/n!](/uploads/image/z/5749832-56-2.jpg?t=%E5%8C%96%E7%AE%801%2F2%21%2B2%2F3%21%2B3%2F4%21%E2%80%A6%E2%80%A6%2Bn-1%2Fn%21)
1/2!+2/3!+3/4!……+n-1/n!
=(2-1)/2!+(3-1)/3!+(4-1)/4!+...+(n-1)/n!
=[2/2!-1/2!]+[3/3!-1/3!]+[4/4!-1/4!]+...+[n/n!-1/n!]
=1-1/2!+1/2!-1/3!+1/3!-1/4!+...+1/(n-1)!-1/n!
=1-1/n!
=(2-1)/2!+(3-1)/3!+(4-1)/4!+...+(n-1)/n!
=[2/2!-1/2!]+[3/3!-1/3!]+[4/4!-1/4!]+...+[n/n!-1/n!]
=1-1/2!+1/2!-1/3!+1/3!-1/4!+...+1/(n-1)!-1/n!
=1-1/n!
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