My question was inspired by this stackexchange question. For the last 90 minutes I have been trying to prove this formula due to Gregorio Fontana:
$$H_n = \gamma + \log n + {1 \over 2n} - \sum_{k=2}^\infty { (k-1)! C_k \over n(n+1)\ldots(n+k-1)}, \qquad \textrm{ for } n=1,2,3,\ldots,$$
where $H_n = \sum\limits_{k=1}^n 1/k$ and the coefficients $C_k$ are the Gregory coefficients given by $${ z \over \log(1-z)} = \sum_{n=0}^\infty C_k z^k \qquad \textrm{ for } |z|<1.$$
It's a bit frustrating as it's something I recall proving as a student many years ago. I have a vague recollection that I began with something like:
$$H_n = \int_0^1 {1-(1-x)^n \over x } \textrm{d}x,$$
but my attempts to follow on from there have failed. Can you help?