tính giá trị : x/1(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+….+ 1/(x+2013)(x+2014) Giúp mình với <<3

Ta để ý rằng

$\dfrac{1}{[x-(n-1)](x-n)} = \dfrac{1}{x-(n-1)} – \dfrac{1}{x-n}$

với mọi số tự nhiên $n$.

Khi đó ta có

$A = \dfrac{x}{x+1} + \dfrac{1}{(x+1)(x+2)} + \dfrac{1}{(x+2)(x+3)} + \cdots + \dfrac{1}{(x+2013)(x+2014)}$

$= \dfrac{x}{x+1} + \dfrac{1}{x+1} – \dfrac{1}{x+2} + \dfrac{1}{x+2} – \dfrac{1}{x+3} + \cdots + \dfrac{1}{x+2013} – \dfrac{1}{x+2014}$

$= \dfrac{x}{x+1} + \dfrac{1}{x+1} – \dfrac{1}{x+2014}$

$= \dfrac{x+1}{x+1} – \dfrac{1}{x+2014} = -\dfrac{1}{x+2014}$