4. Study the convergence of the sequence $\left(x_{n}\right)_{n \geq 1}$, defined by $x_{1}=\sqrt{6}$, and for $n \geq 1$

$$
x_{n+1}=\sqrt{6+x_{n}}
$$
Solution. It is shown inductively that the sequence is increasing $2 \mathrm{p}$

![](https://cdn.mathpix.com/cropped/2024_06_07_e6a8932e6a7b27bb15fag-3.jpg?height=51&width=1106&top_left_y=1470&top_left_x=492)

![](https://cdn.mathpix.com/cropped/2024_06_07_e6a8932e6a7b27bb15fag-3.jpg?height=46&width=1101&top_left_y=1510&top_left_x=495)

Taking the limit in the recurrence relation, we obtain $\lim _{n \rightarrow \infty} x_{n}=3 \ldots \ldots \ldots .2 \mathrm{p}$