If $$\sum_{j=1}^n \frac{x_j}{i+j-1}=1,~ 1\le i \le n,$$ $x_j$ can be found by considering a system of linear simultaneous equations: $\sum_{j=1}^nH_{ij} ~ x_j=1, 1\le i \le n$, where $H_{ij}$ is the Hilbert matrix. See: https://en.wikipedia.org/wiki/Hilbert_matrix The question here is: How to find the simplest form for $x_j$ by this method or otherwise? To re-emphasize, eventually, $x_j$ would be a function of $j$ and $n$ only.
2026-04-07 21:24:10.1775597050
The simplest form for $x_j$ if $\sum_{j=1}^n \frac{x_j}{i+j-1}=1,~ 1\le i \le n$
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