Đặt biểu thức là A
\(\Rightarrow\)A=\(\dfrac{\left(x+1\right)-x}{x\left(x+1\right)}+\dfrac{\left(x+2\right)-\left(x+1\right)}{\left(x+1\right)\left(x+2\right)}+\dfrac{\left(x+3\right)-\left(x+2\right)}{\left(x+2\right)\left(x+3\right)}+...+\dfrac{\left(x+2014\right)-\left(x+2013\right)}{\left(x+2013\right)\left(x+2014\right)}\)
\(\Leftrightarrow\dfrac{x+1}{x\left(x+1\right)}-\dfrac{x}{x\left(x+1\right)}+\dfrac{x+2}{\left(x+1\right)\left(x+2\right)}-\dfrac{x+1}{\left(x+1\right)\left(x+2\right)}+...+\dfrac{x+2014}{\left(x+2013\right)\left(x+2014\right)}-\dfrac{x+2013}{\left(x+2013\right)\left(x+2014\right)}\)\(\Leftrightarrow\dfrac{1}{x}-\dfrac{1}{x+1}+\dfrac{1}{x+1}-\dfrac{1}{x+2}-\dfrac{1}{x+2}-...-\dfrac{1}{x+2013}+\dfrac{1}{x+2013}-\dfrac{1}{x+2014}.\)\(\Leftrightarrow\dfrac{1}{x}-\dfrac{1}{x+2014}\)
\(\Leftrightarrow\dfrac{x+2014-x}{x\left(x+2014\right)}\)
\(\dfrac{2014}{x\left(x+2014\right)}\)
