$ĐKXĐ : x \neq -1,-2,-3,....,-2014$
Ta có :
$\dfrac{1}{x.(x+1)}+\dfrac{1}{(x+1)(x+2)} + ....+\dfrac{1}{(x+2013).(x+2014)} $
$ = \dfrac{1}{x}-\dfrac{1}{x+1}+\dfrac{1}{x+1}-\dfrac{1}{x+2}+....+\dfrac{1}{x+2013}-\dfrac{1}{x+2014}$
$ = \dfrac{1}{x}-\dfrac{1}{x+2014}$
$ = \dfrac{x+2014-x}{x.(x+2014)}$
$ = \dfrac{2014}{x.(x+2014)}$