17. Compute
$$
2 \sqrt{2 \sqrt[3]{2 \sqrt[4]{2 \sqrt[5]{2 \cdots}}}}
$$
Solution: $2^{e-1}$
Taking the base 2 logarithm of the expression gives
$$
1+\frac{1}{2}\left(1+\frac{1}{3}\left(1+\frac{1}{4}(1+\cdots)\right)\right)=1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots=e-1 .
$$

Therefore the expression is just $2^{e-1}$.