<s>[INST] Boundedness of $\frac{1}{n}+\frac{1}{n-x}+\frac{1}{n-2x}+\dots+\frac{1}{n-(n-1)x}$ Let $0\leq x\le 1$, and $n$ a positive integer. Define the function
$$f(n,x)=\frac{1}{n}+\frac{1}{n-x}+\frac{1}{n-2x}+\dots+\frac{1}{n-(n-1)x}.$$
For which $x$ is the sum bounded by a constant for arbitrarily large $n$?
If $x=0$, this is $\frac{1}{n}+\frac{1}{n}+\dots+\frac{1}{n}=1$, a constant. If $x=1$, it is $\frac{1}{n}+\frac{1}{n-1}+\dots+\frac{1}{1}$, a divergent sum. The question is what happens in between.
As mfl wrote, the sum is bounded above by $\frac{1}{1-x}$. Is it also true that the sum converges to $\frac{1}{1-x}$?
 [/INST] Hint
$$\frac{1}{n}+\frac{1}{n-x}+\frac{1}{n-2x}+\dots+\frac{1}{n-(n-1)x}\le  \frac{n}{n-(n-1)x}\underbrace{\nearrow}_{n\to \infty} \frac{1}{1-x}$$ for $0<x<1.$
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