1/(1x3)+1/(3x5)+1/(5x7)+.+1/(2011x2013) 等于什么

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1/(1x3)+1/(3x5)+1/(5x7)+.+1/(2011x2013) 等于什么
1/(1x3)+1/(3x5)+1/(5x7)+.+1/(2011x2013) 等于什么

1/(1x3)+1/(3x5)+1/(5x7)+.+1/(2011x2013) 等于什么
1/(1x3)+1/(3x5)+1/(5x7)+.+1/(2011x2013)
=1/2 x[2/(1x3)+2/(3x5)+2/(5x7)+.+2/(2011x2013)]
=1/2 x[1-1/3+1/3-1/5+1/5-1/7+.+1/2011-1/2013]
=1/2 x[1-1/2013]
=1/2 x[2012/2013]
=1006/2013

1/[(2n-1)(2n+1)]=(1/2)*[1/(2n-1)-1/(2n+1)]

所以1/(1x3)+1/(3x5)+1/(5x7)+....+1/(2011x2013)
=(1/2)*[(1-1/3)+(1/3-1/5)+(1/5-1/7)+...+(1/2011-1/2013)]
=(1/2)*(1-1/2013)=1006/2013