数学1/a(a+1)+1/(a+1)(a+2)+.+1/(a+2009)(a+2010)解法
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数学1/a(a+1)+1/(a+1)(a+2)+.+1/(a+2009)(a+2010)解法
数学1/a(a+1)+1/(a+1)(a+2)+.+1/(a+2009)(a+2010)解法
数学1/a(a+1)+1/(a+1)(a+2)+.+1/(a+2009)(a+2010)解法
1/a(a+1)+1/(a+1)(a+2)+.+1/(a+2009)(a+2010)
采用裂相法
原式=1/a-1/(a+1)+1/(a+1)-1/(a+2)+.1/(a+2009)-1/(a+2010)
中间相相互抵消
=1/a-1/(a+2010)
=2009/a(a+2010)
拆分法
形如1/a(a+1)的分数都可以拆为 1/A - 1/(A+1),所以
原式 = 1/A - 1/(A+1)+1/(A+1)-1/(A+2)+……+1/(A+2009)-1/(A+2010)
= 1/A - 1/(A+2010)
1/a(a+1)=1/a-1/(a+1)
其他的同理
1/a(a+1)+1/(a+1)(a+2)+....+1/(a+2009)(a+2010)
=1/a-1/(a+1)+1/(a+1)-1/(a+2)-……-1/(a+2009)-1/(a+2010)
=1/a-1/(a+2001)=2000/a(a+2001)
这算是比较典型的一种题型
1/a(a+1)+1/(a+1)(a+2)+....+1/(a+2009)(a+2010)
=1/a-1/(a+1)+1/(a+1)-1/(a+2)+...+1/(a+2008)-1/(a+2009)+1/(a+2009)-1/(a+2010)
=1/a-1/(a+2010)
=2010/a(a+2010)