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I had this question bothering me for a while, but I can't come up with a meaningful answer. The problem is the following:

Let integers $a_i,b_j\in${$1,\ldots,n$} and $K_1,K_2\in$ {$1,\ldots,K$}, then how small (as a function of $K$ and $n$), but strictly positive, can the following absolute difference be.

$\biggl|(\sum_{i=1}^{K_1} \frac{1}{a_i})-(\sum_{j=1}^{K_2} \frac{1}{b_j})\biggr|$

As an example for $K_1=1,K_2=1,a_1=n,b_1=n-1$, K_1=1,$$K_2=1$ choosing $a_1=n,$$b_1=n-1$gives the smallest positive absolute difference, that is $\frac{1}{n(n-1)}$. What could the general case be?

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The difference of two sums of unit fractions

I had this question bothering me for a while, but I can't come up with a meaningful answer. The problem is the following:

Let integers $a_i,b_j\in${$1,\ldots,n$} and $K_1,K_2\in$ {$1,\ldots,K$}, then how small (as a function of $K$ and $n$), but strictly positive, can the following absolute difference be.

$\biggl|(\sum_{i=1}^{K_1} \frac{1}{a_i})-(\sum_{j=1}^{K_2} \frac{1}{b_j})\biggr|$

As an example for $K_1=1,K_2=1,a_1=n,b_1=n-1$, the smallest positive absolute difference is $\frac{1}{n(n-1)}$. What could the general case be?