Timeline for Find the tight upper bound of $\sum_{i=1}^n \frac{i}{i+x_i}$, where the $x_i$'s are distinct in $\{1,2,...,n\}$
Current License: CC BY-SA 4.0
3 events
| when toggle format | what | by | license | comment | |
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| Nov 2, 2018 at 16:39 | comment | added | Martin Rubey | In fact, if this pattern persists, then the maximum would be $\frac{1}{4} \, r {\left(H_{n - r/2} - H_{r/2} - 2\right)} + \frac{1}{2} \, n + \frac{r(r+1)}{2 \, {\left(n + 1\right)}}$ for some $r$, where $H_k$ is the harmonic number. | |
| Nov 2, 2018 at 16:17 | comment | added | Martin Rubey | Here are the first few maxima: $([1], 1/2)$, $([1, 2], 1), ([2, 1], 1)$, $([3, 1, 2], 91/60)$, $([4, 1, 2, 3], 214/105)$, $([5, 1, 2, 3, 4], 1613/630)$, $([6, 1, 2, 3, 4, 5], 10679/3465)$, $([7, 1, 2, 3, 4, 5, 6], 1298221/360360)$, $([8, 7, 1, 2, 3, 4, 5, 6], 3469/840)$, $([9, 8, 1, 2, 3, 4, 5, 6, 7], 2609/560)$, $([10, 9, 1, 2, 3, 4, 5, 6, 7, 8], 287579/55440)$ | |
| Nov 2, 2018 at 15:12 | history | answered | js21 | CC BY-SA 4.0 |