@Câu 25. [id1432] (HSG 11 – VĨNH PHÚC 2013-2014) Cho dãy số $\left( {{x}_{n}} \right)$ xác định như sau: $\left\{ \begin{align} & {{x}_{1}}=1 \\ & {{x}_{n+1}}=\sqrt{{{x}_{n}}\left( {{x}_{n}}+1 \right)\left( {{x}_{n}}+2 \right)\left( {{x}_{n}}+3 \right)+1} \\ \end{align} \right.\,\,\,\,\left( n=1,\,2,\,3.... \right)$ Đặt ${{y}_{n}}=\displaystyle\sum\limits_{i=1}^{n}{\dfrac{1}{{{x}_{i}}+2}\,\,\left( n=1,\,2,\,3....... \right)}$. Tính $\underset{x\to +\infty }{\mathop{\lim }}\,{{y}_{n}}$

@Câu 25. [id1432] (HSG 11 – VĨNH PHÚC 2013-2014) Cho dãy số $\left( {{x}_{n}} \right)$ xác định như sau: $\left\{ \begin{align} & {{x}_{1}}=1 \\ & {{x}_{n+1}}=\sqrt{{{x}_{n}}\left( {{x}_{n}}+1 \right)\left( {{x}_{n}}+2 \right)\left( {{x}_{n}}+3 \right)+1} \\ \end{align} \right.\,\,\,\,\left( n=1,\,2,\,3.... \right)$ Đặt ${{y}_{n}}=\displaystyle\sum\limits_{i=1}^{n}{\dfrac{1}{{{x}_{i}}+2}\,\,\left( n=1,\,2,\,3....... \right)}$. Tính $\underset{x\to +\infty }{\mathop{\lim }}\,{{y}_{n}}$