TIFR-2019-Maths-A: 13

Question

Let the sequence $\left \{ x_{n} \right \}_{n\rightarrow 1}^{\infty }$ be defined by $x1=\sqrt{2}$ and $x_{n+1}=\left ( \sqrt{2} \right )^{x_{n}}$ for  $n\geq 1$. Then which one of the following statements is true?

• A. The sequence $\left \{ x_{n} \right \}$ is monotonically increasing and  $\underset{n\rightarrow \infty }{lim}\:x_{n}=2$
• B. The sequence $\left \{ x_{n} \right \}$ is neither monotonically increasing nor monotonically decreasing
• C. $\underset{n\rightarrow \infty }{lim}\:x_{n}$ does not exist
• D. $\underset{n\rightarrow \infty }{lim}\:x_{n}=\infty$