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关于前段时间所谓“韦神”出的“难题”

2023-04-01 12:20 作者:げいしも_芸  | 我要投稿

先看题

我们直接开始解答

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%3D%5Csum_%7B1%5Cleq%20i%2Cj%5Cleq%20n%7D%5Cleft%20(%5Csum_%7Bk_1%3D1%7D%5E%7B%2B%5Cinfty%7D(-1)%5E%7Bk_1%2B1%7D%5Cfrac%7B(a_ia_j)%5E%7B2k_1%2B1%7D%7D%7B2k_1%2B1%7D%2B%5Csum_%7Bk_2%3D1%7D%5E%7B%2B%5Cinfty%7D(-1)%5E%7Bk_2%2B1%7D%5Cfrac%7B(-a_ia_j)%5E%7B2k_2%2B1%7D%7D%7B2k_2%2B1%7D%5Cright)

%3D2%5Csum_%7Bk%3D0%7D%5E%7B%2B%5Cinfty%7D%5Csum_%7B1%5Cleq%20i%2Cj%5Cleq%20n%7D%5Cfrac%7B(a_ia_j)%5E%7B2k%2B1%7D%7D%7B2k%2B1%7D

%3D%5Csum_%7Bk%3D0%7D%5E%7B%2B%5Cinfty%7D%5Cleft(%5Cfrac%7B2%7D%7B2k%2B1%7D%5Csum_%7Bi%3D1%7D%5Ena_i%5E%7B2k%2B1%7D%5Csum_%7Bj%3D1%7D%5Ena_j%5E%7B2k%2B1%7D%5Cright)

%3D%5Csum_%7Bk%3D0%7D%5E%7B%2B%5Cinfty%7D%5Cfrac%202%7B2k%2B1%7D%5Cleft(%5Csum_%7Bi%3D1%7D%5Ena_i%5E%7B2k%2B1%7D%5Cright)%5E2%5Cgeq0

从最后的式子不难看出取等的条件为

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在这里给出其成立的一个必要条件,读者可自证充分性

记a_1,a_2,a_3...a_n为数列{a_n},则

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