Interesting Limit (24)

$H_n$ is defined as follows:

$\begin{aligned} H_n & = \frac n{\frac 1{\binom n0} + \frac 1{\binom n1} + \frac 1{\binom n2} + \cdots+ \frac 1{\binom nn}} = \frac n{1+\frac 1n + \frac 2{n(n-1)} + \cdots + \frac 2{n(n-1)} + \frac 1n + 1} < \frac n2 & \small \color{#3D99F6} \text{for }n \ge 2 \end{aligned}$

This implies that $H_n \sim \dfrac n2$ as $n \to \infty$ and

$\begin{aligned} \lim_{n \to \infty} \sqrt[n]{H_n} & = \lim_{n \to \infty} \left(\frac n2\right)^{\frac 1n} \\ & = \lim_{n \to \infty} \exp \left(\frac 1n (\ln n - \ln 2)\right) & \small \color{#3D99F6} \text{where }\exp (x) = e^x \\ & = \exp \left(\color{#3D99F6} \lim_{n \to \infty} \frac {\ln n - \ln 2}n \right) & \small \color{#3D99F6} \text{A }\infty / \infty \text{ case, L'Hôpital's rule applies.} \\ & = \exp \left(\lim_{n \to \infty} \frac {\frac 1n}1 \right) & \small \color{#3D99F6} \text{Differentiate up and down w.r.t. }n. \\ & = e^0 = \boxed 1 \end{aligned}$