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首先搞懂1/[x*(x+1)]=1/x-1/(x+1)所以原式=1-1/2+1/2-1/3+1/3-1/4.....-1/2008+1/2008-1/2009=1-1/2009=2008/2009...

千问 | 2009-8-5 13:09:49 | 显示全部楼层

1/2+1/2*3+1/3*4+1/4*5+......+1/2008*2009=1/2+（1/2-1/3）+(1/3-1/4)+（1/4-1/5）+...+（1/2008-1/2009）=1/2+1/2-1/2009=1-1/2009=2008/2009...

千问 | 2009-8-5 13:09:49 | 显示全部楼层

1/(1*2)+1/(2*3)+1/(3*4)+^^^=(1/1-1/2)+(1/2-1/3)+(1/3-1/4)+^^^+(1/2007-1/2008)+(1/2008-1/2009)=1-1/2009=2008/2009...

千问 | 2009-8-5 13:09:49 | 显示全部楼层

1/(n*(n+1))=1/n-1/(n+1)所以上式=1-1/2+1/2-1/3+......+1/2008-1/2009=1-1/2009=2008/2009...

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