Radicals Overload

Calculus Level 3

$\large 2 \displaystyle\sqrt{2\sqrt[3]{2 \sqrt[4]{2 \sqrt[5]{2 \cdots}}}} = \,?$

Clarification: $\displaystyle e = \lim_{n\to\infty} \left(1 + \dfrac1n\right)^n \approx 2.718$ .