1/1*2+1/2*3+1/3*4+1/4*5+.+1/2006*2007

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1/1*2+1/2*3+1/3*4+1/4*5+.+1/2006*2007
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1/1*2+1/2*3+1/3*4+1/4*5+.+1/2006*2007
1/1*2+1/2*3+1/3*4+1/4*5+.+1/2006*2007

1/1*2+1/2*3+1/3*4+1/4*5+.+1/2006*2007
1/(1×2)+1/(2×3)+1/(3×4)+1/(4×5)+……+1/(2006×2007)
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+……+(1/2005-1/2006)+(1/2006-1/2007)
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+……+1/2005-1/2006+1/2006-1/2007
=1-1/2007
=2006/2007