Is it true that $\sum_{i=1}^n \prod_{j \neq i} \frac{1}{x_j-x_i} = 0$ for distinct $x's$? I tried for $n=1,2,3,4,5$ and these are true. But I cannot generalize it for natural number $n$. I just tried mathematical induction, but it was also difficult. Anyone know this equation? (I don't know what tags are going with, so if you have a good idea, change it please.)
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This needs some clarification - which values of j are considered in the product? All natural numbers excluding i? – BallBoy May 25 '22 at 15:54
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j's are 1, 2, ... , n except for i. – Han Jeong May 25 '22 at 15:55