根据上述方法计算:1/1*3+1/3*5+1/5*7+.....+1/2 005*2007

显示全部楼层 · 2012-10-27 16:39:08

解：数列:1/1*3+1/3*5+1/5*7+.....+1/(2n-1)*（2n+1）1/(2n-1)*（2n+1）=1/2×[1/(2n-1）-1/(2n+1)]∴1/1*3=1/2×[1-1/3]1/3*5=1/2×[1/3-1/5]……1/(2n-1)*（2n+1）=1/2×[1/(2n-1）-1/(2n+1)]1/1*3+1/3*5+1/5*7+.....+1/2 005*2007=1/2×[1-1/3]+1/2×[1/3-1/5]+……+1/2×[1/2005-1/2007]=1/2×（1-1/3+1/3-1/5+……+1/2005-1/2007）=1/2×（1-1/2007）=1/2×2006/...

千问 · 2012-10-27 16:39:08

1/1*3+1/3*5+1/5*7+.....+1/2 005*2007=（1-1/3+1/3-1/5+1/5-1/7+……+1/2005-1/2007）÷2=（1-1/2007）÷2=2006/2007÷2=1003/2007...

千问 · 2012-10-27 16:39:08

由裂项公式1/(n(n+d))=1/d(1/n-1/(n+d))1/1*3+1/3*5+1/5*7+.....+1/2 005*2007=1/2（1/1-1/3+1/3-1/5+1/5-1/7+……+1/2005-1/2007)中间项消掉=1/2(1-1/2007)=1003/2007...