Timeline for Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$

7 events

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Apr 12 at 17:25	comment	added	LSpice		@RichardStanley, re, doesn't @‍FedorPetrov's answer address that? (Based on the timestamps, it might have appeared slightly after your comment.)
Apr 12 at 16:19	comment	added	Pietro Majer		@RichardStanley I would conjecture that the equality of the series for a set of exponents $k$ with $\sum 1/k=\infty$, implies the equality for all $k\ge1$
Apr 11 at 19:26	comment	added	Richard Stanley		Yes, I wasn't criticizing your answer but just emphasizing that it doesn't answer the original question.
Apr 11 at 18:56	history	edited	Max Alekseyev	CC BY-SA 4.0	added 164 characters in body
Apr 11 at 18:32	comment	added	Max Alekseyev		$x_i$ being positive is an assumption, while as I put in the disclaimer my answer is assumption-free. Surely, assumptions like this one make the question harder.
Apr 11 at 18:27	comment	added	Richard Stanley		The question remains whether we can find positive real sequences $x_1,x_2,\dots$ and $y_1,y_2,\dots$ such that $\sum_i x_i^{2k}=\sum_i y_i^{2k}$ for all $k\geq 1$, but $\sum x_i\neq \sum y_i$; or even more strongly, $\sum_i x_i^k=\sum_i y_i^k$ for all $k\geq 2$, but $\sum x_i\neq \sum y_i$.
Apr 11 at 17:56	history	answered	Max Alekseyev	CC BY-SA 4.0