Question

1/(1*2)+1/(2*3)+......+1/(n*(n+1))=1-1/(n+1)，求推导过程

Answer 1

1/(1*2)=1-1/2
1/(2*3)=1/2-1/3
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1/(n*(n+1))=1/(n-1)-1/(n+1)
左右两边同时相加,得
1/(1*2)+1/(2*3)+......+1/(n*(n+1))=1-1/2+1/2-1/3+1/3........+1/(n-1)-1/(n+1)=1-1/(n+1)
这个叫裂项法.

Answer 2

1/(1*2)+1/(2*3)+......+1/(n*(n+1))
=1-1/2+1/2-1/3+1/3-1/4.....+1/(n-1)-1/n+1/n-1/(n+1)
中间的项全消去了,只剩下两 边的数:
=1-1/(n+1)

Answer 3

原式=(1-1/2)+(1/2-1/3)+…+(1/(n-1)-1/n)+(1/n-1/(n+1))
=1-1/(n+1)