(1-1/2^)(1-1/3^)……(1-1/2007^)(1-1/2008^)

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(1-1/2^)(1-1/3^)……(1-1/2007^)(1-1/2008^)
(1-1/2^)(1-1/3^)……(1-1/2007^)(1-1/2008^)

(1-1/2^)(1-1/3^)……(1-1/2007^)(1-1/2008^)
用平方差
原式=(1-1/2)(1+1/2)(1-1/3)(1+1/3)……(1-1/2008)(1+1/2008)
=(1/2)(3/2)(2/3)(4/3)……(2007/2008)(2009/2008)
中间约分
=(1/2)(2009/2008)
=2009/4016

(1-1/2^)(1-1/3^)……(1-1/2007^)(1-1/2008^)
=1/2*3/2*2/3*4/3*....*2006/2007*2008/2007*2007/2008*2009/2008
=1/2*2009/2008
=2009/4016
先用平方差

2008
(1-1/2009)=------
2009
2007
(1-1/2008)=------
2008
...
2
(1-1/3) = -----
...

全部展开

2008
(1-1/2009)=------
2009
2007
(1-1/2008)=------
2008
...
2
(1-1/3) = -----
3
1
(1-1/2) = -----
2
所以
(1-1/2009)(1-1/2008)(1-1/2007)……(1-1/3)(1-1/2)
2008 2007 2006 2 1
= ---- ---- ---- ... --- ---
2009 2008 2007 3 2
挫项相消
1
= ----
2009

收起

???啥意思说明白点

2009/4016

S=(1-1/2^)(1-1/3^)……(1-1/2007^)(1-1/2008^)
=(1/2)*(2/3)*(3/4)*(4/5)...(2006/2007)*(2007/2008)
=1/2008(前一个分母和后一个分子恰好可以相乘为一,剩下后第一个分子和最后一个分母)
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