原文链接:
具体数学-第4课 - WeiYang Blog
今天讲了多重求和,也就是一个和式由多个下标来指定。
首先是最简单的形式:![\sum\limits_{1 \le j,k \le n} { {a_j}{b_k}} = (\sum\limits_{1 \le j \le n} { {a_j}} )(\sum\limits_{1 \le k \le n} { {a_k}} )](https://www.zhihu.com/equation?tex=%5Csum%5Climits_%7B1+%5Cle+j%2Ck+%5Cle+n%7D+%7B+%7Ba_j%7D%7Bb_k%7D%7D+%3D+%28%5Csum%5Climits_%7B1+%5Cle+j+%5Cle+n%7D+%7B+%7Ba_j%7D%7D+%29%28%5Csum%5Climits_%7B1+%5Cle+k+%5Cle+n%7D+%7B+%7Ba_k%7D%7D+%29)
例题1
下面给出一个对称矩阵: 求: 这是这个矩阵的上三角加对角线求和,因为是对称的嘛,可以补全下三角,加上对角线就行了。 所以![S = \frac{1}{2}({(\sum\limits_{1 \le k \le n} { {a_k}} )^2} + \sum\limits_{1 \le k \le n}^{} { {a_k}^2} )](https://www.zhihu.com/equation?tex=S+%3D+%5Cfrac%7B1%7D%7B2%7D%28%7B%28%5Csum%5Climits_%7B1+%5Cle+k+%5Cle+n%7D+%7B+%7Ba_k%7D%7D+%29%5E2%7D+%2B+%5Csum%5Climits_%7B1+%5Cle+k+%5Cle+n%7D%5E%7B%7D+%7B+%7Ba_k%7D%5E2%7D+%29)
例题2
下面再看一个例子: 同样模仿上例调换 位置,得到: 所以 至此解完,然后可以推出一个著名的不等式————切比雪夫不等式: 如果 那么 反之如果 那么 更一般的结论,给定两个序列 和 ,求下面式子最大值与最小值: 其中 是 的一个排列。答案是 增序最大,降序最小,至于为什么,下面给出两种证明方法。
方法1
如上图所示, 和 按照递增顺序排列,每个方格的面积代表 与 的乘积,记为 。那么上面的求和式其实就是每一行每一列都必须有且只有一块被取。考虑第一行,如果不取 ,取其他的 ,那么第一列也只能取其他的 ,这样的话 也就取不了了。但是发现 并且两种取法影响的行和列都是相同的,这说明了,取 和 不如取 和 。所以 必取,然后第一行第一列就不能取了。剩下的方阵用相同的方法可以得出必取 ,也就是主对角线。同理最小取法用副对角线可以推出。
方法2
设数列 和 非单调递减,那么有如下证明: 反之亦证。
题外话,其实切比雪夫不等式原来是以微积分形式给出的:如果函数 和 非单调递减,那么有:![(\int_a^b {f(x)dx} )(\int_a^b {g(x)dx} ) \le (b - a)(\int_a^b {f(x)g(x)dx} )](https://www.zhihu.com/equation?tex=%28%5Cint_a%5Eb+%7Bf%28x%29dx%7D+%29%28%5Cint_a%5Eb+%7Bg%28x%29dx%7D+%29+%5Cle+%28b+-+a%29%28%5Cint_a%5Eb+%7Bf%28x%29g%28x%29dx%7D+%29)
例题3
求 我将用三种方法来求解这个式子。
方法1
首先将 和 分开,首先计算对 求和:![\begin{array}{l}S = \sum\limits_{1 \le k \le n} {\sum\limits_{1 \le j k} {\frac{1}{ {k - j}}} } \\ = \sum\limits_{1 \le k \le n} {\sum\limits_{1 \le k - j k} {\frac{1}{j}} } \\ = \sum\limits_{1 \le k \le n} {\sum\limits_{0 j \le k - 1} {\frac{1}{j}} } \\ = \sum\limits_{1 \le k \le n} { {H_{k - 1}}} \\ = \sum\limits_{0 \le k n} { {H_k}} \end{array}](https://www.zhihu.com/equation?tex=%5Cbegin%7Barray%7D%7Bl%7DS+%3D+%5Csum%5Climits_%7B1+%5Cle+k+%5Cle+n%7D+%7B%5Csum%5Climits_%7B1+%5Cle+j+%3C+k%7D+%7B%5Cfrac%7B1%7D%7B+%7Bk+-+j%7D%7D%7D+%7D+%5C%5C+%3D+%5Csum%5Climits_%7B1+%5Cle+k+%5Cle+n%7D+%7B%5Csum%5Climits_%7B1+%5Cle+k+-+j+%3C+k%7D+%7B%5Cfrac%7B1%7D%7Bj%7D%7D+%7D+%5C%5C+%3D+%5Csum%5Climits_%7B1+%5Cle+k+%5Cle+n%7D+%7B%5Csum%5Climits_%7B0+%3C+j+%5Cle+k+-+1%7D+%7B%5Cfrac%7B1%7D%7Bj%7D%7D+%7D+%5C%5C+%3D+%5Csum%5Climits_%7B1+%5Cle+k+%5Cle+n%7D+%7B+%7BH_%7Bk+-+1%7D%7D%7D+%5C%5C+%3D+%5Csum%5Climits_%7B0+%5Cle+k+%3C+n%7D+%7B+%7BH_k%7D%7D+%5Cend%7Barray%7D)
方法2
先计算对 求和:![\begin{array}{l}S = \sum\limits_{1 \le j \le n} {\sum\limits_{j k \le n} {\frac{1}{ {k - j}}} } \\ = \sum\limits_{1 \le j \le n} {\sum\limits_{j k + j \le n} {\frac{1}{k}} } \\ = \sum\limits_{1 \le j \le n} {\sum\limits_{0 k \le n - j} {\frac{1}{k}} } \\ = \sum\limits_{1 \le j \le n} { {H_{n - j}}} \\ = \sum\limits_{0 \le j n} { {H_j}} \end{array}](https://www.zhihu.com/equation?tex=%5Cbegin%7Barray%7D%7Bl%7DS+%3D+%5Csum%5Climits_%7B1+%5Cle+j+%5Cle+n%7D+%7B%5Csum%5Climits_%7Bj+%3C+k+%5Cle+n%7D+%7B%5Cfrac%7B1%7D%7B+%7Bk+-+j%7D%7D%7D+%7D+%5C%5C+%3D+%5Csum%5Climits_%7B1+%5Cle+j+%5Cle+n%7D+%7B%5Csum%5Climits_%7Bj+%3C+k+%2B+j+%5Cle+n%7D+%7B%5Cfrac%7B1%7D%7Bk%7D%7D+%7D+%5C%5C+%3D+%5Csum%5Climits_%7B1+%5Cle+j+%5Cle+n%7D+%7B%5Csum%5Climits_%7B0+%3C+k+%5Cle+n+-+j%7D+%7B%5Cfrac%7B1%7D%7Bk%7D%7D+%7D+%5C%5C+%3D+%5Csum%5Climits_%7B1+%5Cle+j+%5Cle+n%7D+%7B+%7BH_%7Bn+-+j%7D%7D%7D+%5C%5C+%3D+%5Csum%5Climits_%7B0+%5Cle+j+%3C+n%7D+%7B+%7BH_j%7D%7D+%5Cend%7Barray%7D)
方法3
按对角线求和:![\begin{array}{l}S = \sum\limits_{1 \le j k \le n} {\frac{1}{ {k - j}}} \\ = \sum\limits_{1 \le j k + j \le n} {\frac{1}{k}} \\ = \sum\limits_{1 \le k \le n} {\sum\limits_{1 \le j \le n - k} {\frac{1}{k}} } \\ = \sum\limits_{1 \le k \le n} {\frac{ {n - k}}{k}} \\ = n\sum\limits_{1 \le k \le n} {\frac{1}{k} - } \sum\limits_{1 \le k \le n} 1 \\ = n{H_n} - n\end{array}](https://www.zhihu.com/equation?tex=%5Cbegin%7Barray%7D%7Bl%7DS+%3D+%5Csum%5Climits_%7B1+%5Cle+j+%3C+k+%5Cle+n%7D+%7B%5Cfrac%7B1%7D%7B+%7Bk+-+j%7D%7D%7D+%5C%5C+%3D+%5Csum%5Climits_%7B1+%5Cle+j+%3C+k+%2B+j+%5Cle+n%7D+%7B%5Cfrac%7B1%7D%7Bk%7D%7D+%5C%5C+%3D+%5Csum%5Climits_%7B1+%5Cle+k+%5Cle+n%7D+%7B%5Csum%5Climits_%7B1+%5Cle+j+%5Cle+n+-+k%7D+%7B%5Cfrac%7B1%7D%7Bk%7D%7D+%7D+%5C%5C+%3D+%5Csum%5Climits_%7B1+%5Cle+k+%5Cle+n%7D+%7B%5Cfrac%7B+%7Bn+-+k%7D%7D%7Bk%7D%7D+%5C%5C+%3D+n%5Csum%5Climits_%7B1+%5Cle+k+%5Cle+n%7D+%7B%5Cfrac%7B1%7D%7Bk%7D+-+%7D+%5Csum%5Climits_%7B1+%5Cle+k+%5Cle+n%7D+1+%5C%5C+%3D+n%7BH_n%7D+-+n%5Cend%7Barray%7D)
由此得到了一个完全不同的表示形式!所以我们得到了:![\sum\limits_{0 \le j n} { {H_j}} = n{H_n} - n](https://www.zhihu.com/equation?tex=%5Csum%5Climits_%7B0+%5Cle+j+%3C+n%7D+%7B+%7BH_j%7D%7D+%3D+n%7BH_n%7D+-+n)
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