Tính giới hạn lim(1-2/2.3)(1-2/3.4)..(1-2/(n+1)(n+2))

* Hướng dẫn giải

Đặt  

 $$\begin{gathered} x_n=\left(1-\frac{2}{2.3}\right)\left(1-\frac{2}{3.4}\right) \\ ....\left(1-\frac{2}{\left(n+1\right)\left(n+2\right)}\right) \end{gathered}$$

Từ

$$\begin{gathered} 1-\frac{2}{\left(k+1\right)\left(k+2\right)} \\ =\frac{k\left(k+3\right)}{\left(k+1\right)\left(k+2\right)} \\ k=1,...,n \end{gathered}$$

ta có

$$\begin{gathered} x_n=\frac{1.4}{2.3}\frac{2.5}{3.4}\frac{3.6}{4.5} \\ ...\frac{n\left(n+3\right)}{\left(n+1\right)\left(n+2\right)} \\ =\frac{n+3}{3\left(n+1\right)} \end{gathered}$$ 

Vậy $lim{x}_n=\frac{1}{3}$

Đáp án C