2/3*1+2/5*3+2/7*5+---+2/2005*2003

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2/3*1+2/5*3+2/7*5+---+2/2005*20032/3*1+2/5*3+2/7*5+---+2/2005*20032/3*1+2/5*3+2/7*5+---+2/2005*2003裂

2/3*1+2/5*3+2/7*5+---+2/2005*2003
2/3*1+2/5*3+2/7*5+---+2/2005*2003

2/3*1+2/5*3+2/7*5+---+2/2005*2003
裂项:
2/3*1+2/5*3+2/7*5+---+2/2005*2003
=2(1/3+1/5*3+...+1/2005*2003)
=2*1/2(1-1/3+1/3-1/5+...+1/2003-1/2005)
=1-1/2005
=2004/2005

2/3*1+2/5*3+2/7*5+---+2/2005*2003
=2*(1/1*3+1/3*5+1/5*7+...+1/2003*2005)
=2*1/2*(1-1/3+1/3-1/5+1/5-1/7+...+1/2003-1/2005)
=1-1/2005
=2004/2005