计算:1乘3/1+3乘5/1……2007乘2009/1
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![计算:1乘3/1+3乘5/1……2007乘2009/1](/uploads/image/z/214912-64-2.jpg?t=%E8%AE%A1%E7%AE%97%3A1%E4%B9%983%2F1%2B3%E4%B9%985%2F1%E2%80%A6%E2%80%A62007%E4%B9%982009%2F1)
计算:1乘3/1+3乘5/1……2007乘2009/1
计算:1乘3/1+3乘5/1……2007乘2009/1
计算:1乘3/1+3乘5/1……2007乘2009/1
1/(1*3)+1/(3*5)+.+1/(2007*2009)
=1/2*(1-1/3)+1/2*(1/3-1/5)+.+1/2*(1/2007-1/2009)
=1/2*(1-1/3+1/3-1/5+.+1/2007-1/2009)
=1/2*(1-1/2009)
=1/2*2008/2009
=1004/2009
1乘3/1+3乘5/1……2007乘2009/1
=2分之1乘(1-3/1+3/1-5/1+...+2007/1-2009/1)
=2分之1×(1-2009/1)
=2分之1×2009分之2008
=2009分之1004
分数不是这样表示的,应该是1/1*3+1/3*5吧
1/1*3+1/3*5+1/5*7+1/7*9+......1/2007*2009
=1/2*[1/1-1/3+1/3-1/5+1/5-1/7+1/7-1/9.....+1/2007-1/2009]
=1/2*(1-1/2009)
=1/2*2008/2009
=1004/2009
利用公式:1/[(2n-1)(2n+1)]=1/2[1/(2n-1)-1/(2n+1)],
原式等于n=1,2,3,4......1004的和的值。
所以:1/(1*3)+1/(3*5)+......+1/(2007*2009)
=1/2*(1-1/3)+1/2*(1/3-1/5)+......+1/2*(1/2007-1/2009)
=1/2*(1-1/3+1/3-1/5+......+1/2007-1/2009)
=1/2*(1-1/2009)
=1/2*(2008/2009)
=1004/2009