Calculus
# Limits of Sequences and Series

Tetration is defined as

$\Large {^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{n \ a\text{'s}}.$

Find the value of

$\lim_{n\rightarrow\infty}{^{n}}\big(\sqrt {2}\big).$

$\lim_{n\to\infty}\frac{P_{n-1} P_{n+1}}{P_{n}^2} =$

The sequence $\{a_n\}$ follows the recursion $a^2_{n+1}=2a_n+3$ with $a_1=7.$

Determine $\displaystyle \lim_{n \to \infty} a_n$.