1/（1*3）+1/（3*5）+1/（5*7）+···+1/（2009*2011）=1/2(1-1/3)+1/2(1/3-1/5)+1/2(1/5-1/7)+.+1/2(1/2009-1/2011)=1/2(1-1/3+1/3-1/5+1/5-1/7+.+1/2009-1/2011)=1/2(1-1/2011)=1/2 * 2010/2011=1005/2011...