令\(x_k=\sqrt{k}y_k\)
則\(\displaystyle \sum_{k=1}^n ky_k^2+2 \sum_{1 \le k<j \le n}ky_k y_j=1\)
令\(\cases{y_1+y_2+\ldots+y_n=a_1 \cr y_2+\ldots+y_n=a_2 \cr \ldots \cr y_n=a_n}\)
則\(a_1^2+a_2^2+\ldots+a_n^2=1\)
令\(a_{n+1}=0\)
則
\(\displaystyle \sum_{i=1}^n x_i\)
\(\displaystyle =\sum_{k=1}^n x_k\)
\(\displaystyle =\sum_{k=1}^n \sqrt{k}y_k\)
\(\displaystyle =\sum_{k=1}^n \sqrt{k}(a_k-a_{k+1})\)
\(\displaystyle =\sum_{k=1}^n \sqrt{k}a_k-\sum_{k=1}^n \sqrt{k}a_{k+1}\)
\(\displaystyle =\sum_{k=1}^n \sqrt{k}a_k-\sum_{k=1}^n \sqrt{k-1}a_k\)
\(\displaystyle =\sum_{k=1}^n (\sqrt{k}-\sqrt{k-1})a_k\)
\(\displaystyle \le \sqrt{\sum_{k=1}^n (\sqrt{k}-\sqrt{k-1})^2}\times \sqrt{\sum_{k=1}^n a_k^2}\)
\(\displaystyle =\sqrt{\sum_{k=1}^n (\sqrt{k}-\sqrt{k-1})^2}\)
