观察 1/1*2+1/2*3==(1-1/2)+(1/2-1/3)=1-1/3=2/3 ,计算2/(x+1)(x+3)+2/(x+3)(x+5)+...+2/(x+2003)(x+2005)观察 1/1*2+1/2*3=(1-1/2)+(1/2-1/3)=1-1/3=2/3 , 计算2/(x+1)(x+3)+2/(x+3)(x+5)+...+2/(x+2003)(x+2005)

观察 1/1*2+1/2*3==(1-1/2)+(1/2-1/3)=1-1/3=2/3 ,计算2/(x+1)(x+3)+2/(x+3)(x+5)+...+2/(x+2003)(x+2005)观察 1/1*2+1/2*3=(1-1/2)+(1/2-1/3)=1-1/3=2/3 , 计算2/(x+1)(x+3)+2/(x+3)(x+5)+...+2/(x+2003)(x+2005)
观察 1/1*2+1/2*3==(1-1/2)+(1/2-1/3)=1-1/3=2/3 ,计算2/(x+1)(x+3)+2/(x+3)(x+5)+...+2/(x+2003)(x+2005)
观察 1/1*2+1/2*3=(1-1/2)+(1/2-1/3)=1-1/3=2/3 , 计算2/(x+1)(x+3)+2/(x+3)(x+5)+...+2/(x+2003)(x+2005)

观察 1/1*2+1/2*3==(1-1/2)+(1/2-1/3)=1-1/3=2/3 ,计算2/(x+1)(x+3)+2/(x+3)(x+5)+...+2/(x+2003)(x+2005)观察 1/1*2+1/2*3=(1-1/2)+(1/2-1/3)=1-1/3=2/3 , 计算2/(x+1)(x+3)+2/(x+3)(x+5)+...+2/(x+2003)(x+2005)
2/(x+1)(x+3)=1/(x+1)-1/(x+3).
2/(x+3)(x+5)=1/(x+3)-1/(x+5).
2/(x+5)(x+7)=1/(x+5)-1/(x+7).
.
2/(x+2001)(x+2003)=1/(x+2001)-1/(x+2003).
2/(x+2003)(x+2005)=1/(x+2003)-1/(x+2005).
将以上的全部相加,
则可以得出答案：1/(x+1)-1/(x+2005).
即2004/(x+1)(x+2005)

=2*(1/(x+1)-1/(x+3)+(1/x+3)-1/(x+5)+...+(1/x+2003)-1/(x+2005))=2*(1/(x+1)-1/(x+2005))
=(4008/(x+1)*(x+2005)

2004/(x+1)*(x+2005)