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1/(x+1)(x+2)+1/(x+2)(x+3)+1/(x+3)(x+4)+.+1/(x+2012)(x+2013)
题目详情
1/(x+1)(x+2)+1/(x+2)(x+3)+1/(x+3)(x+4)+.+1/(x+2012)(x+2013)
▼优质解答
答案和解析
1/(x+1)(x+2)+1/(x+2)(x+3)+1/(x+3)(x+4)+.+1/(x+2012)(x+2013)
=[1/(x+1)-1/(x+2)]+[1/(x+2)-1/(x+3)]+[1/(x+3)-1/(x+4)]+.+[1/(x+2002)-1/(2003)]
=1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+1/(x+3)-1/(x+4)+.+1/(x+2002)-1/(x+2003)
中间的所有项刚好正负抵消,只剩前后两项
=1/(x+1)-1/(x+2003)
=(x+2003)/[(x+1)(x+2003)]-(x+1)/[(x+1)(x+2003)]
=2002/[(x+1)(x+2003)]
=[1/(x+1)-1/(x+2)]+[1/(x+2)-1/(x+3)]+[1/(x+3)-1/(x+4)]+.+[1/(x+2002)-1/(2003)]
=1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+1/(x+3)-1/(x+4)+.+1/(x+2002)-1/(x+2003)
中间的所有项刚好正负抵消,只剩前后两项
=1/(x+1)-1/(x+2003)
=(x+2003)/[(x+1)(x+2003)]-(x+1)/[(x+1)(x+2003)]
=2002/[(x+1)(x+2003)]
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