1/2*4+1/4*6+1/6*8+.+1/2006*2008=?
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1/2*4+1/4*6+1/6*8+.+1/2006*2008=?
1/2*4+1/4*6+1/6*8+.+1/2006*2008=?
1/2*4+1/4*6+1/6*8+.+1/2006*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(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
=1/4*(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*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