Binomial coefficient inverses

Compute. $n=2$: $\frac{1}{1}-\frac{1}{2}+\frac{1}{1}=1.5$.

Conclusion. For $n=2$, the sum is $3/2$.

General. $\sum_{k=0}^{n}\frac{(-1)^k}{\binom{n}{k}}=\frac{(-1)^n\cdot n+1}{n+1}\cdot\frac{1}{?}$ — the general formula is more complex.