原式=1/(1x2)+1/(2x3)+1/(3x4)+1/(4X5)+1/(5x6)+1/(6x7)+1/(7x8)
=(1/1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7)+(1/7-1/8)
注：1/(n(n+1))=1/n-1/(n+1)
=1-1/8
=7/8
这是数列求和中很常用的处理手段
1/2+1/6+1/12+1/20+1/30+1/42+1/56
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7)+(1/7-1/8)
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8
=1-1/8
=7/8
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