Tính giới hạn: 

- Hướng dẫn giải

Đáp án là A

$lim\left[\frac{1}{1.3}+\frac{1}{2.4}+...+\frac{1}{n\left(n+2\right)}\right]$

Ta có :

 $$\begin{gathered} lim\left[\frac{1}{1.3}+\frac{1}{2.4}+...+\frac{1}{n\left(n+2\right)}\right] \\ =lim\frac{1}{2}.\left[\frac{2}{1.3}+\frac{2}{2.4}+...+\frac{2}{n\left(n+2\right)}\right] \end{gathered}$$

$$\begin{gathered} lim\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{5}...+\frac{1}{n}-\frac{1}{n+2}\right) \\ =lim\frac{1}{2}\left(1+\frac{1}{2}-\frac{1}{n+2}\right)= \frac{1}{2}(1\hspace{0.17em}+\hspace{0.17em}\frac{1}{2}-0)=\frac{3}{4} \end{gathered}$$