By induction on \(n\) it is shown that \[ \sum_{k=0}^n \binom{n}{k}^{-1} = \frac{(n+1)}{2^{n+1}} \sum_{k=1}^{n+1} \frac{2^k}{k} \] and \[ \sum_{k=0}^\infty (-1)^k \binom{n+k-1}{k} = \frac{n}{2}\left(2^n \ln 2 - \sum_{k=1}^{n-1}\frac{2^k}{k} \frac{2^k}{n-k}\right) \] hold. The special case \(\displaystyle\sum_k \binom{n+k-1}{k} = \frac{n}{n - 1}\) of Lerch’s theorem is also proved.