Đặt A=\(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{x.\left(x+1\right)}\)
A=\(\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{x}-\dfrac{1}{x+1}\)
A=1-\(\dfrac{1}{x+1}\)
A= \(\dfrac{x+1}{x+1}\) - \(\dfrac{1}{x+1}\)
A=\(\dfrac{x}{x+1}\)
\(A=\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{x\left(x+1\right)}\)
\(A=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{x}-\dfrac{1}{x+1}\)
\(A=1-\dfrac{1}{x+1}\)
\(A=\dfrac{x+1}{x+1}-\dfrac{1}{x+1}=\dfrac{x+1-1}{x+1}=\dfrac{x}{x+1}\)