Sum of the reciprocals of the binomial coefficients (Q685302)

Let \(S_ n=\sum^ n_{k=0}{1 \over {n \choose k}}\). It is shown that \(S_ n\) satisfies the recurrence \(S_ n={n+1 \over 2n}S_{n-1}+1\). The proof can be simplified by observing that \[ n!S_ n=\sum^ n_{k=0}k!(n-k)!=n!+\sum^{n-1}_{k=0}k!(n-k-1)!(n+1-k-1) =n!+(n+ 1)(n-1)!S_{n-1}-(n!S_ n-n!). \]