TITLE: What is the asymptotic behavior of A103213 in OEIS?
QUESTION [9 upvotes]: It's probably not at all hard—but at least right now it's not obvious to me—how to determine the asymptotic behavior of
$\sum_{k=1}^n \binom{n}{k} \frac{1}{k}$
(link to OEIS).

REPLY [5 votes]: Here is an exact lower bound, which, as is readily seen, approximately equal to $2^{n+1}/n$.
By Jensen's inequality, since $1/x$ is convex, 
$$
\frac{{\sum\nolimits_{k = 1}^n {{n \choose k}} }}{{\sum\nolimits_{k = 1}^n {{n \choose k}} k}} \leq \frac{{\sum\nolimits_{k = 1}^n {{n \choose k}\frac{1}{k}} }}{{\sum\nolimits_{k = 1}^n {{n \choose k}} }}.
$$
From this it follows straightforwardly that
$$
\frac{{(2^n  - 1)^2 }}{{2^{n - 1} n}} \le \sum\limits_{k = 1}^n {{n \choose k}\frac{1}{k}} .
$$
EDIT: Hence,
$$
\sum\limits_{k = 1}^n {{n \choose k}\frac{1}{k}} \geq \frac{{2^{n + 1} }}{n} - \frac{4}{n} + \frac{1}{{2^{n - 1} n}}.
$$