2/3+2/8+2/15+2/24+.+2/2008*2010这道题如何算

2/3+2/8+2/15+2/24+.+2/2008*2010这道题如何算
2/3+2/8+2/15+2/24+.+2/2008*2010这道题如何算

2/3+2/8+2/15+2/24+.+2/2008*2010这道题如何算
原式=2/(1*3)+2/(2*4)+2/(3*5)+2/(4*6)+...+2/(2007*2009) + 2/(2008*2010)
裂项有 =(1-1/3)+(1/2-1/4)+(1/3-1/5)+(1/4-1/6)+...+(1/2007-1/2009)+(1/2008-1/2010)
抵消中间项=1 + 1/2 -1/2009-1/2010
=3/2-1/2009-1/2010 = 754/503

一般项为2/(n*(n+2))=1/n-1/(n+2)，所以其和为
1-1/3+1/2-1/4+1/3-1/5+...+1/2006-1/2008+1/2007-1/2009+1/2008-1/2010
=1+1/2-1/2009-1/2010=...
最后结果不用算了吧

原式=2/(1*3)+2/(2*4)+2/(3*5)+2/(4*6)+...+2/(2007*2009) + 2/(2008*2010)
裂项有 =(1-1/3)+(1/2-1/4)+(1/3-1/5)+(1/4-1/6)+...+(1/2007-1/2009)+(1/2008-1/2010)
抵消中间项=1 + 1/2 -1/2009-1/2010
=3/2-1/2009-1/2010 = 754/503