Asymptotic expansion of harmonic sum constants

Let $f(n) = \sum_{k=1}^{n} \frac{1}{k}$. Then there exist constants $\gamma$, $c$, and $d$ such that $$ f(n) = \ln(n) + \gamma + \frac{c}{n} + \frac{d}{n^{2}} + O\left(\frac{1}{n^{3}}\right) $$ where the $O\left(\frac{1}{n^{3}}\right)$ means terms of order $\frac{1}{n^{3}}$ or lower. Compute the ordered pair $(c, d)$.