被 EI 教育了

\begin{align}
\sum_{l=1}^n \frac{1}{l}H_{n-l} &= \sum_{l=1}^n \frac{1}{l} \sum_{j=1}^{n-l} \frac{1}{j}
\\ &= \sum_{i + j \leq n} \frac{1}{ij}
\\ &= \sum_{i+j \leq n} \frac{1}{i+j}\left(\frac{1}{i} + \frac{1}{j}\right)
\\ &= 2\sum_{k \leq n} \frac{1}{k} \sum_{i < k} \frac{1}{i}
\\ &= 2 \sum_{k \leq n} \frac{H_{k-1}}{k} = 2A.
\\ A &= \sum_{k \leq n} \frac{H_{k-1}}{k}
\\ &= \sum_{i=1}^{n} \frac{1}{i} \sum_{i < j \leq n} \frac{1}{j}
\\ &= \sum_{i=1}^{n} \frac{1}{i} \left(H_n - H_i\right)
\\ &= H_n^2 - \sum_{i=1}^n \frac{H_i}{i}
\\ &= H_n^2 - H_n^{(2)} - A.
\end{align}
为啥 align 后面必须加文字才能渲染出来啊,这是什么新锅

posted @ 2022-03-08 00:07  cjoier_Itst  阅读(1521)  评论(2编辑  收藏  举报