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计算1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3)+···+1/(a+2004)(a+2005)

分类: 作业答案 • 2022-01-02 16:04:35
问题描述：

计算1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3)+···+1/(a+2004)(a+2005)

答

1/n(n+1)=1/n-1/(n+1)
1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3)+···+1/(a+2004)(a+2005)
=1/a-1/(a+1)+1/(a+1)-1/(a+2)+...+1/(a+2004)-1/(a+2005)
=1/a-1/(a+2005)

答

1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3)+···+1/(a+2004)(a+2005)
=1/a-1/(a+1)+1/(a+1)-1/(a+2)+1/(a+2)-1/(a+3)+---+1/(a+2004)-1/(a+2005)
=1/a-1/(a+2005)
=2005/(a²+2005a)

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