TITLE: Asymptotic rate for $\sum\binom{n}k^{-1}$
QUESTION [5 upvotes]: This MO question prompted me to ask:

What is the second order asymptotic growth/decay rate for the sum
  $$\sum_{k=0}^n\frac1{\binom{n}k}$$
  as $n\rightarrow\infty$?

REPLY [5 votes]: The following asymptotic expansion is proved in https://arxiv.org/abs/0904.1757 (The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, by Nicholas Pippenger. Published in  Mathematics Magazine 83(N5) (2010), 331-346):
$$\sum_{k=0}^n\frac{1}{\binom{n}{k}}=2\left(1+\frac{1}{n}+\frac{2}{n(n-1)}+\ldots \frac{k!}{n(n-1)\cdots(n-k+1)}+O(\frac{1}{n^{k+1}})\right).$$
P.S. This result confirms the answer given in the Gerhard Paseman's comment.