This [MO question](https://mathoverflow.net/questions/301903/what-is-the-value-of-this-double-sum-in-closed-form) 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 $n\rightarrow\infty$?
> 
> 
> 



Asymptotic rate for $\sum\binom{n}k^{-1}$
This sum can be written in the form, see [2-adic Logarithm and Resistance of n-dimensional Cube](https://mathoverflow.net/questions/146285/2-adic-logarithm-and-resistance-of-n-dimensional-cube)
$$S\_{n+1}=\frac{n+1}{2^{n+1}}\sum\_{k=1}^{n+1}\frac{2^k}{k}.$$
The last term in the sum gives the estimate $S\_n>1.$ You can get sharper estimates taking more terms.


