(1-1/2*1/2)*(1-1/3*1/3)*(1-1/4*1/4)...*(1-1/2005*1/2005)怎样简便计算

(1-1/2*1/2)*(1-1/3*1/3)*(1-1/4*1/4)...*(1-1/2005*1/2005)怎样简便计算
(1-1/2*1/2)*(1-1/3*1/3)*(1-1/4*1/4)...*(1-1/2005*1/2005)怎样简便计算

(1-1/2*1/2)*(1-1/3*1/3)*(1-1/4*1/4)...*(1-1/2005*1/2005)怎样简便计算
原式=[(2*2-1)/(2*2)]*[(3*3-1)/(3*3)]……[(2005*2005-1)/(2005*2005)]
=[(2-1)*(2+1)/(2*2)]*[(3-1)*(3+1)/(3*3)]……[(2005-1)*(2005+1)/(2005*2005)]
=[(1*3) /(2*2)]*[(2*4)/(3*3)]……[(2004*2006)/(2005*2005)]
=(1/2)*(2006/2005)
=1003/2005

=(1+1/2)(1-1/2)(1+1/3)(1-1/3)...(1+1/2005)(1-1/2005)
=3/2*1/2*4/3*2/3*5/4*3/4...2006/2005*2004/2005
=3/2*4/3*5/4...2006/2005*1/2*2/3*3/4...*2004/2005(上式奇偶位交换下）
=2006/2*1/2005
=2006/4010

原式=[(2*2-1)/(2*2)]*[(3*3-1)/(3*3)]……[(2005*2005-1)/(2005*2005)]
=[(2-1)*(2+1)/(2*2)]*[(3-1)*(3+1)/(3*3)]……[(2005-1)*(2005+1)/(2005*2005)]
=[(1*3) /(2*2)]*[(2*4)/(3*3)]……[(2004*2006)/(2005*2005)]
=(1/2)*(2006/2005)
=1003/2005

(1-1/2*1/2)*(1-1/3*1/3)*(1-1/4*1/4)...*(1-1/2005*1/2005)
=(1+1/2)*(1-1/2)*(1+1/3)*(1-1/3)*(1+1/4)*(1-1/4)....*(1+1/2004)*(1-1/2004)*(1+1/2005)*(1-1/2005)
=(1+1/2)*（1+1/3)*(1+1/4).....*(1+1/200...

全部展开

(1-1/2*1/2)*(1-1/3*1/3)*(1-1/4*1/4)...*(1-1/2005*1/2005)
=(1+1/2)*(1-1/2)*(1+1/3)*(1-1/3)*(1+1/4)*(1-1/4)....*(1+1/2004)*(1-1/2004)*(1+1/2005)*(1-1/2005)
=(1+1/2)*（1+1/3)*(1+1/4).....*(1+1/2004)*(1+1/2005)*(1-1/2)*(1-1/3)*(1-1/4)....*(1-1/2004)*(1-1/2005)
=3/2*4/3*5/4....*2005/2004*2006/2005*1/2*2/3*3/4.....*2003/2004*2004/2005
互相约分得到=2006/2*1/2005=1003/2005
相信我，我的才是正确答案！

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