Return to Revisions

[incomplete] Assuming $x_i\ge0$ and $y_i\ge0$ with $ \sum_i x_i <\infty$, $\sum y_i <\infty$ and $\sum_i x_i^k=\sum y_i^k$ for all $k> N$ we have that $\phi(t):=\sum_i(e^{x_it}-1)=\sum_{k\ge1} \big(\sum_i x_i^k\big)t^k/k!$ and $\psi(t):=\sum_i(e^{y_it}-1)$ are entire functions (we can expand the exponentials and exchange order of summation, by absolute summability). It remains to prove that $\phi(t) -\psi(t) =o(|t|)$ for $t\to-\infty$; then since $\psi(t)-\phi(t)=\sum_{k=1}^N(\sum_i x_i^k - \sum_i y_i^k)t^k/k!$ is a polynomial, it would follow it is identically zero, so $\sum_i x_i^k=\sum_i y_i^k$ for $1\le k\le N$ too.