I can't figure this out, can someone help me prove it? I know you guys will come up with an incredibly elegant solution
2026-04-04 16:48:38.1775321318
How can I prove $\sum_{i=1}^n \sum_{j=1}^n \frac{a_ia_j}{i+j-1}$ is never negative for any set of n real numbers $a_i$
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Hint: Since $\dfrac{1}{i+j-1} = \displaystyle\int_{0}^{1}x^{i+j-2}\,dx$, we have $$\sum_{i = 1}^{n}\sum_{j = 1}^{n}\dfrac{a_ia_j}{i+j-1} = \sum_{i = 1}^{n}\sum_{j = 1}^{n}\int_{0}^{1}a_ia_jx^{i+j-2}\,dx = \int_{0}^{1}\sum_{i = 1}^{n}a_ix^{i-1} \cdot \sum_{j = 1}^{n}a_jx^{j-1}\,dx$$
Can you see why this is non-negative?