Help is needed for a limit question $\lim_{n\rightarrow\infty}\sum^{n}_{k=1}\frac{1}{ \binom{n}{k}}$

Question

$$\lim_{n\rightarrow\infty}\sum^{n}_{k=1}\frac{1}{ \binom{n}{k}}$$

Answer 1

for Lower bound

Let $$ a_{n} = \sum^{n}_{k=1}\frac{1}{\binom{n}{k}} = \frac{1}{\binom{n}{1}}+\sum^{n-2}_{k=2}\frac{1}{\binom{n}{k}}+\frac{1}{\binom{n}{n-1}}+\frac{1}{\binom{n}{n}}\geq 1+\frac{2}{n}$$

and for upper bound $$a_n=\sum_{k=1}^{n}\frac{1}{\binom{n}{k}}.$$ Then $$a_n=1+\frac{2}{n}+ \sum_{k=2}^{n-2}\frac{1}{\binom{n}{k}}\le 1+\frac{2}{n}+\frac{n-3}{\binom{n}{2}}=1+\frac{4}{n}.\frac{n-2}{n-1}$$

Now $n\rightarrow \infty$ and $$\lim_{n\to \infty} \sum_{k=1}^{n}\frac{1}{\binom{n}{k}}=1.$$