Abel变换

$%5Cdisplaystyle%20%7B%5Csum%5E%7Bn%7D_%7Bi%3D1%7D%7D(a_i%20b_i)%3D%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7D%5Cleft%20(%5Csum%5E%7Bi%7D_%7Bj%3D1%7D%5Cleft%20(%20(b_%7Bj%7D%5Cright%20)%20(a_%7Bi%7D-a_%7Bi%2B1%7D))%20%5Cright%20)%20%2B%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7D%5Cleft%20(b_%7Bi%7D%20a_%7Bn%7D%20%5Cright%20)$

Abel变换的证明

Abel变换的代数证明

$%5Cbegin%7Balign%7D%20%26%5Cdisplaystyle%20%7B%5Csum%5E%7Bn%7D_%7Bi%3D1%7D%7D(a_i%20b_i)%20%5C%5C%20%3D%26%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7D%20%5Cleft%20((a_i)%5Cleft%20(%20-%5Csum%5E%7Bi-1%7D_%7Bj%3D1%7Db_j%20%2B%20%5Csum%5E%7Bi%7D_%7Bj%3D1%7Db_j%5Cright%20)%5Cright)%20%5C%5C%20%3D%26%20%2B%20a_1%20%5Ccolor%7Bblue%7D%7B%5Csum%5E%7B1%7D_%7Bj%3D1%7Db_j%7D%20-%20a_2%20%5Ccolor%7Bblue%7D%7B%5Csum%5E%7B1%7D_%7Bj%3D1%7Db_j%7D%20%5C%5C%26%20%2B%20a_2%20%5Ccolor%7Bred%7D%7B%5Csum%5E%7B2%7D_%7Bi%3D1%7Db_j%7D%20-%20a_3%20%5Ccolor%7Bred%7D%7B%5Csum%5E%7B2%7D_%7Bi%3D1%7Db_j%7D%20%20%5C%5C%20%26%2B%5Ccdots%20%20%5C%5C%20%26%2B%20a_%7Bn-2%7D%20%5Ccolor%7Bgreen%7D%7B%5Csum%5E%7Bn-2%7D_%7Bi%3D1%7Db_%7Bj%7D%7D-%20a_%7Bn-1%7D%20%5Ccolor%7Bgreen%7D%7B%5Csum%5E%7Bn-2%7D_%7Bi%3D1%7Db_%7Bj%7D%7D%20%5C%5C%20%26%2B%20a_%7Bn-1%7D%20%5Ccolor%7Bpurple%7D%7B%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7Db_%7Bj%7D%7D%20-%20a_%7Bn%7D%20%5Ccolor%7Bpurple%7D%7B%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7Db_%7Bj%7D%7D%20%5C%5C%20%26%20%2B%20a_%7Bn%7D%20%5Ccolor%7Bbrown%7D%7B%7B%5Csum%5E%7Bn%7D_%7Bi%3D1%7Db_%7Bj%7D%7D%7D%20%5C%5C%20%3D%26%2B%20(a_1%20-%20a_2)%20%5Ccolor%7Bblue%7D%7B%5Csum%5E%7B1%7D_%7Bj%3D1%7Db_j%7D%20%20%5C%5C%26%2B(a_2%20-%20a_3)%20%5Ccolor%7Bred%7D%7B%5Csum%5E%7B2%7D_%7Bj%3D1%7Db_j%7D%20%20%5C%5C%26%2B%20%5Ccdots%20%20%5C%5C%26%2B%20(a_%7Bn-2%7D%20-%20a_%7Bn-3%7D)%20%5Ccolor%7Bgreen%7D%7B%5Csum%5E%7Bn-2%7D_%7Bj%3D1%7Db_j%7D%20%5C%5C%26%2B%20(a_%7Bn-1%7D%20-%20a_%7Bn-2%7D)%20%5Ccolor%7Bpurple%7D%7B%5Csum%5E%7Bn-1%7D_%7Bj%3D1%7Db_j%7D%20%5C%5C%26%2B%20a_%7Bn%7D%20%5Ccolor%7Bbrown%7D%7B%7B%5Csum%5E%7Bn%7D_%7Bi%3D1%7Db_%7Bj%7D%7D%7D%20%20%5C%5C%3D%26%20%2B%20%5Csum%5E%7B1%7D_%7Bj%3D1%7D%5Cleft%20(b_j(a_1%20-%20a_2)%20%5Cright%20)%20%5C%5C%26%2B%20%5Csum%5E%7B2%7D_%7Bj%3D1%7D%5Cleft%20(%20b_j(a_2%20-%20a_3)%20%5Cright%20)%20%5C%5C%26%2B%20%5Ccdots%20%20%5C%5C%26%2B%20%5Csum%5E%7Bn-2%7D_%7Bj%3D1%7D%5Cleft%20(%20b_j(a_%7Bn-2%7D%20-%20a_%7Bn-3%7D)%5Cright%20)%20%5C%5C%26%2B%20%5Csum%5E%7Bn-1%7D_%7Bj%3D1%7D%5Cleft%20(%20b_j(a_%7Bn-1%7D%20-%20a_%7Bn-2%7D)%5Cright%20)%20%5C%5C%26%2B%20%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7D(b_%7Bj%7Da_%7Bn%7D)%20%5C%5C%3D%26%20%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7D%5Cleft%20(%5Csum%5E%7Bi%7D_%7Bj%3D1%7D%5Cleft%20(%20(b_%7Bj%7D%5Cright%20)%20(a_%7Bi%7D-a_%7Bi%2B1%7D))%20%5Cright%20)%20%2B%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7D%5Cleft%20(b_%7Bi%7D%20a_%7Bn%7D%20%5Cright%20)%5Cend%7Balign%7D$

Abel变换的几何证明

天才“阿贝尔”的灵感：梯形面积计算，直观地理解“阿贝尔公式” —— 来自UP主：究尽数学

习题

请证明Abel变换
请运用Abel变换来证明全体自然数的平方和公式:
asdfasdfasdf

答案

$1.$

$%5Cbegin%7Balign%7D%202.%5Cquad%20%261%5E2%2B2%5E2%2B%20%5Ccdots%20%2B%20(n-1)%5E2%20%2Bn%5E2%20%3D%20%5Cdisplaystyle%20%5Csum%20%5E%7Bn%7D_%7Bi%3D1%7Di%5E2%20%5C%5C%3D%26%20%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7D%5Cleft%20(%5Csum%5E%7Bi%7D_%7Bj%3D1%7Dj(i-(i%2B1))%20%5Cright%20)%20%2B%20n%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7Di%20%5C%5C%3D%26-%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7D%5Cleft%20(%20%5Csum%5E%7Bi%7D_%7Bj%3D1%7Dj%20%5Cright%20)%2B%20n%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7Di%20%20%5C%5C%3D%26%20-%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7D%5Cleft%20(%5Cfrac%7B(1%2Bi)i%7D%7B2%7D%5Cright%20)%20%20%2B%20n%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7Di%20%5C%5C%3D%26%20-%5Cfrac%7B%5Cdisplaystyle%20%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7Di%5E2%2B%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7Di%7D%7B2%7D%2Bn%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7Di%20%20%5C%5C%3D%26%20-%5Cfrac%7B%5Cdisplaystyle%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7Di%5E2%2B%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7Di-n%5E2%7D%7B2%7D%2Bn%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7Di%20%20%5C%5C%5Cimplies%20%26%20%20%5Csum%20%5E%7Bn%7D_%7Bi%3D1%7Di%5E2%3D%20%20-%5Cfrac%7B%5Cdisplaystyle%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7Di%5E2%2B%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7Di-n%5E2%7D%7B2%7D%2Bn%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7Di%20%20%5C%5C%20%5Cimplies%20%26%203%5Csum%20%5E%7Bn%7D_%7Bi%3D1%7Di%5E2%3D%20%20-%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7Di%2Bn%5E2%2B2n%20%5Csum%5E%7Bn%7D_%7Bi%3D1%7Di%20%20%5C%5C%20%5Cimplies%20%26%20%5Csum%20%5E%7Bn%7D_%7Bi%3D1%7Di%5E2%3D%5Cfrac%7B-%5Cfrac%7Bn(n-1)%7D%7B2%7D%2Bn%5E2%2B(n%2B1)n%5E2%7D%7B3%7D%20%5C%5C%20%5Cimplies%20%26%20%5Csum%20%5E%7Bn%7D_%7Bi%3D1%7Di%5E2%3D%5Cfrac%7B-n%5E2%2Bn%2B2n%5E2%2B2n%5E3%2B2n%5E2%7D%7B6%7D%20%5C%5C%20%5Cimplies%20%26%20%5Csum%20%5E%7Bn%7D_%7Bi%3D1%7Di%5E2%3D%5Cfrac%7B2n%5E3%2B3n%5E2%2Bn%7D%7B6%7D%20%5Cend%7Balign%7D$

注：若你在解第2题的过程中做了如下的变换：

$-%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7D%5Cleft%20(%20%5Csum%5E%7Bi%7D_%7Bj%3D1%7Dj%20%5Cright%20)%3D-%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7Di%5E2%2Bn%5Csum%5E%7Bn-1%7D_%7Bi%3D1%7Di$

则你将会回到起点。

$3.%20$

搬运部分

天才“阿贝尔”的灵感：梯形面积计算，直观地理解“阿贝尔公式” —— 来自UP主：究尽数学

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