Ta có : \(\sqrt{1+\dfrac{1}{a}+\dfrac{1}{\left(a+1\right)^2}}=\sqrt{\left(1+\dfrac{1}{a}\right)^2-\dfrac{2}{a}+\dfrac{1}{\left(a+1\right)^2}}=\sqrt{\left(\dfrac{1+a}{a}\right)^2-2.\dfrac{a+1}{a}.\dfrac{1}{a+1}+\dfrac{1}{\left(a+1\right)^2}}=\sqrt{\left(1+\dfrac{1}{a}-\dfrac{1}{a+1}\right)^2}=\text{ |}1+\dfrac{1}{a}-\dfrac{1}{a+1}\text{ |}\) Áp dụng điều này vào bài toán , ta có :