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求和：1/2!+2/3!+3/4!+...+n/(n+1)!
题目详情
求和：1/2!+2/3!+3/4!+...+n/(n+1)!
▼优质解答
答案和解析
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)!,
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)!,
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