Is the following true?
$$
\sum_{i=1}^n \frac{s_i}{i + \sqrt{s_i}} = O( \sqrt{\sum_{i=1}^n s_i} \log n )
$$
where $s_i \geq 1, \forall i$.
2026-03-28 14:06:36.1774706796
Is it true that $ \sum_{i=1}^n \frac{s_i}{i + \sqrt{s_i}} = O( \sqrt{\sum_{i=1}^n s_i} \log n ) $?
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No. Take $s_i=i$. Then
$$\sum_{i=1}^n\frac{i}{i+\sqrt{i}}\approx\int_1^n\frac{x}{x+\sqrt{x}}\,dx=1-2\sqrt{n}+n-\log 4+2\log(1+\sqrt{n})=O(n)$$ whereas the right hand side is $O(n\log n)$.