求和:1/2!+2/3!+3/4!+...+n/(n+1)!

来源:学生作业帮助网 编辑:作业帮 时间:2024/04/29 16:33:20
求和:1/2!+2/3!+3/4!+...+n/(n+1)!

求和:1/2!+2/3!+3/4!+...+n/(n+1)!
求和:1/2!+2/3!+3/4!+...+n/(n+1)!

求和:1/2!+2/3!+3/4!+...+n/(n+1)!
用裂项法求和
n/(n+1)!=[(n+1)-1] /(n+1)!
=(n+1) /(n+1)!-1 /(n+1)!
=1/ n!-1 /(n+1)!.
1/2!+2/3!+3/4!+...+n/(n+1)!
=[1/1!-1 /2!]+[ 1/2!-1 /3!]+[ 1/3!-1 /4!]+...+[ 1/ n!-1 /(n+1)!]
=1-1 /(n+1)!,

原式=(1-1/2!)+(1/2!-1/3!)+(1/3!-1/4!)+...+[1/(n-1)!-1/n!]+[1/n!-1/(n+1)!]
=1-1/(n+1)!