化简x(x+1)分之1+(x+1)(x+2)分之1+(x+2)(x+3)分之1+.+(x+2010)(x+2011)分之一,并求当x=1时的值

问题描述:

化简x(x+1)分之1+(x+1)(x+2)分之1+(x+2)(x+3)分之1+.+(x+2010)(x+2011)分之一,并求当x=1时的值

1/[x(x+1)]+1/[(x+1)(x+2)]+1/[(x+2)(x+3)]+...+1/[(x+2011)(x+2012)]
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+...+1/(x+2011)-1/(x+2012) 中间的都消掉了
=1/x-1/(x+2012)
=(x+2012-x)/[x(x+2012)]
=2012/[x(x+2012)]
当x=1时
原式=2012/2013

x(x+1)分之1+(x+1)(x+2)分之1+(x+2)(x+3)分之1+.+(x+2010)(x+2011)分之一=1/x-1/(x+1)+1/(x+1)-1/(x+2)+……+1/(x+2010)-1/(x+2011)=1/x-1/(x+2011)=2011/x(x+2011)当x=1时原式=2011/2012