Is there any way to express $\sum_{n=0}^{x-2} \frac {1}{x-n} $ as a function of $x$, as opposed to being a summation? I tried doing this through modelling on Desmos, but the closest I could get was $y= \ln(\lfloor{x+0.5}\rfloor)-\ln(2)+\frac{2}{3}$ which isn't even that close to the original function.
If it is possible to express it as a function, how would one get to it, and what would it be?
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If $x > 2$ is not an integer, I presume you mean the sum to go up to $\lfloor x-2 \rfloor$ (i.e. the sum over all $n$ such that $0 \le n \le x-2$). Then your sum can be written as $\psi(-x) - \psi(\lfloor x\rfloor - x - 1)$ where $\psi$ is the digamma function. For integer $x \ge 2$ you get $H_x - 1$ as Yves Daoust wrote.
If $x$ is a natural, the sum is $H_x-1$ where $H_n$ denotes the $n^{th}$ harmonic number.