数学题请给我详细过程 1⼀2*4+1⼀4*6+1⼀6*8+......+1⼀2006*2008=?

=1/2（1/2-1/4）+1/2（1/4-1/6）+1/2（1/6-1/8）+....+1/2（1/2006-1/2008）
=1/2*（1/2-1/4+1/4-1/6+1/6-1/8+........+1/2006-1/2008）
=1/2*（1/2-1/2008）
=1/2*1003/2008
=1003/4016

=1/2(1/2-1/4+1/4-1/6+1/6-1/8+...+1/2006-1/2008)
=1/2(1/2-1/2008)
=1003/4016

此类问题皆可以下列方法求解
1/A*B = 1/(B-A)*(1/A-1/B)

1/2*4+1/4*6+1/6*8+......+1/2006*2008
=1/4*(1/1*2+1/2*3+...+1/1003*1004)
=1/4*[(1/1-1/2)+(1/2-1/3)+...+(1/1003-1/1004)]
=1/4*(1-1/1004)
=1003/4016

=1/2(1/2-1/4)+1/2(1/4-1/6)+……1/2(1/2006-1/2008)
=1/2(1/2-1/2008)
=1003/4016

裂项

1/2(1/2-1/4+1/4-1/6+...+1/2006-1/2008)=1/2(1/2-1/2008)=1/2*1003/2008=1003/4016

1/2*4+1/4*6+1/6*8+.......+1/2006*2008
=(1/2-1/4+1/4-1/6+1/6-1/8+......+1/2006-1/2008)*1/2
=1003/2008*1/2
=1003/4016