> 
> Consider the following limit:
>  $$\lim\_{n \to \infty} \sum\_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}{\binom{n-k}{k}\frac{1}{2^{n-k}}}.$$
>  I can find the limit numerically, but is it possible to compute it analytically?
> 
> 
> 

The answer is $\frac{2}{3}$.