计算:{1\2005}+{1/2004-1\2003}+{1/2003-1/2002}+……+{1/3-1/2}+{1/2-1}

来源：百度知道 编辑：UC知道 时间：2024/06/25 06:44:58
计算:{1\2005}+{1/2004-1\2003}+{1/2003-1/2002}+……+{1/3-1/2}+{1/2-1}
请列出详细步骤，并加以说明分析！
注意：“{ }”表示绝对值， “/”是分数线。

因为1/2004<1/2003,故{1/2004-1/2003}＝1/2003-1/2004
其他同理
{1\2005}+{1/2004-1\2003}+{1/2003-1/2002}+……+{1/3-1/2}+{1/2-1}
＝1/2005+1/2003-1/2004+1/2002-1/2003+……+1-1/2
＝1/2005-1/2004+1

{1\2005}+{1/2004-1\2003}+{1/2003-1/2002}+……+{1/3-1/2}+{1/2-1}

{1\2005}+{1/2004-1\2003}+{1/2003-1/2002}+……+{1/3-1/2}+{1/2-1}
=1/2005+1/2003-1/2004+1/2002-1/2003+……+1/2-1/3+1-1/2
=1/2005-1/2004+1
=1/2005+2003/2004
=4018019/4018020

因为1/N+1<1/N,所以{1/N+1-1/N}=1/N-1/N+1
于是，原式变为:1/2005+1/2003-1/2004+1/2002-1/2003+1/2001-1/2002.....+1/2-1/3+1-1/2
=1/2005-1/2004+1