计算:1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+.+1/(x+2008)(x+2009)

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计算:1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+.+1/(x+2008)(x+2009)

1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+.+1/(x+2008)(x+2009)
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+.+1/(x+2008)-1/(x+2009)
=1/x-1/(x+2009)
=[(x+2009)-x] / [x(x+2009)]
=2009 / [x(x+2009)]