Recurring Styles #2 - Recurrence on Higher Degrees!

Calculus Level 5

$\Large{\frac{a_n}{a_{n-1}} = \sqrt[6]{\frac{a_{n-1}}{a^2_{n-2}}}}$

Define a sequence $a_n$ that satisfies the recurrence relation as described above where $n \geq 2 , a_0 = e , a_1 = e^2$ .

Find the value of $\displaystyle\large{ \lim_{n \to \infty} a_n}$ .

Bonus: Generalize for $a_n$ .