By the well-known properties of the forward difference operator,
if $f$ is a polynomial with degree $\leq 6$ we have $$ \sum_{k=0}^{7}\binom{7}{k}(-1)^k\,f(r+k) = 0 $$ for any $r\in\mathbb{R}$. In our case, by picking $r=0$ and exploiting the binomial theorem we get $$ f(0)=\sum_{k=1}^{7}\binom{7}{k}\frac{(-1)^{k+1}}{k} =\int_{0}^{1}\frac{1-(1-x)^7}{x}\,dx=\int_{0}^{1}\frac{x^7-1}{x-1}\,dx=H_7=\color{red}{\frac{363}{140}}.$$ The approach suggested by Lord Shark the Unknown in the comments leads to $$ f(0)=\left.\frac{d}{dx}(x-1)\cdots(x-7)\right|_{x=0}\stackrel{\text{LogDerivative}}{=} H_7$$ the same outcome, which is not surprising.