Approximation of $2^n$ for large $n$

Question

Is there some sort of approximation for $2^n$? I'm specifically interested in how the ratio $\frac{2n}{2^n}$ scales for large $n$ (apart from decreasing to zero in the limit)

Answer 1

Let $r_n = \frac{2n}{2^n}$. Then $$ \frac{r_{n+1}}{r_n}=\frac12\left(1+\frac1n\right) \to \frac12 $$ In words, for large $n$, we have that $r_{n+1}$ is approximately half of $r_n$ but slightly greater.