6. [5] Determine the value of $\lim _{n \rightarrow \infty} \sum_{k=0}^{n}\binom{n}{k}^{-1}$.
Answer: 2 Let $S_{n}$ denote the sum in the limit. For $n \geq 1$, we have $S_{n} \geq\binom{ n}{0}^{-1}+\binom{n}{n}^{-1}=2$. On the other hand, for $n \geq 3$, we have
$$
S_{n}=\binom{n}{0}^{-1}+\binom{n}{1}^{-1}+\binom{n}{n-1}^{-1}+\binom{n}{n}^{-1}+\sum_{k=2}^{n-2}\binom{n}{k}^{-1} \leq 2+\frac{2}{n}+(n-3)\binom{n}{2}^{-1}
$$
which goes to 2 as $n \rightarrow \infty$. Therefore, $S_{n} \rightarrow 2$.