解:1/x(x+1) + 1/(x+1)(x+2) + ...... +1/(x+2010)(x+2011)
注意:1/x(x+1)=(x+1-x)/x(x+1)=(x+1)/x(x+1)-x/x(x+1)=1/x-1/(x+1)
原式 =1/x -1/(x+1) + 1/(x+1)- 1/(x+2)+........+1/(x+2010) - 1/(x+2011)
=1/x-1/(x+2011)
=2010/x(x+2011)
1/x(x+1)=1/x-(1/1+x)
1/(1+x)(2+x)=(1/1+x)-1/(2+X)
合项相加得2010/(x+2011)
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)+……+1/(x+2010)-1/(x+2011) =2011/[x(x+2011)]
1/x(x+1) + 1/(x+1)(x+2) + ...... +1/(x+2010)(x+2011)
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+........+1/(x+2010)-1/(x+2011)
=1/x-1/(x+2011)
=2010/x(x+2011)