简便计算1/6+1/12+1/20+1/30+.+1/2005*2006=

简便计算1/6+1/12+1/20+1/30+.+1/2005*2006=
简便计算1/6+1/12+1/20+1/30+.+1/2005*2006=

简便计算1/6+1/12+1/20+1/30+.+1/2005*2006=
1/6+1/12+1/20+1/30+.+1/2005*2006=
=[1/(2*3)]+[1/(3*4)]+[1/(4*5)]+[1/(5*6)]+……+[1/(2005*2006)]
=(1/2)-(1/3)+(1/3)-(1/4)+(1/4)-(1/5)+(1/5)-(1/6)+……+(1/2005)-(1/2006)
=(1/2)-(1/2006)
=1002/2006
=501/1003

1/6+1/12+1/20+1/30+.......+1/2005*2006=
(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+......+(1/2005-1/2006)
=1/2-1/2006=1002/2006=501/1003

原式= 1/(2*3) + 1/(3*4) +...+1/(2005*2006)
= 1/2 - 1/3 + 1/3 - 1/4 + ... +1/2005 - 1/2006
= 1/2 - 1/2006
= 501/1003

501/1003