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1十1/（1十2）十1/（1＋2十3）十…十1/（1十2十3十…十100）的值
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1十1/（1十2）十1/（1＋2十3）十…十1/（1十2十3十…十100）的值
▼优质解答
答案和解析
1+2+……+n=n(n+1)/2
1/(1+2+……+n)=2/{n(n+1)}=2*[(1/n)-1/(n+1)}
所以1+1/(1+2)+1/(1+2+3)+……+1/(1+2+3+……+100)
=2*[(1/1-(1/2))+((1/2)-(1/3))+((1/3)-(1/4))……+((1/100)-(1/101))]
=2*(1/1-(1/101))
=200/101
1/(1+2+……+n)=2/{n(n+1)}=2*[(1/n)-1/(n+1)}
所以1+1/(1+2)+1/(1+2+3)+……+1/(1+2+3+……+100)
=2*[(1/1-(1/2))+((1/2)-(1/3))+((1/3)-(1/4))……+((1/100)-(1/101))]
=2*(1/1-(1/101))
=200/101
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