Tính giới hạn lim (1/3+1/6+1/10+...+2/(n+1)(n+2), n thuộc N sao

* Hướng dẫn giải

$$\begin{gathered} \lim \left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{(\mathrm{n}+1)(\mathrm{n}+2)}\right) \\ =\lim \left[2\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{(\mathrm{n}+1)(\mathrm{n}+2)}\right)\right] \\ =\lim [2.(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+... \\ +\frac{1}{\mathrm{n}+1}-\frac{1}{\mathrm{n}+2}\left)\right] \\ =\lim \left[2.\left(\frac{1}{2}-\frac{1}{\mathrm{n}+2}\right)\right]=1 \end{gathered}$$