杂项

Abel’s lemma

$$\begin{aligned} \sum_{k=1}^n a_k b_k &= a_1 b_1 + \sum_{k=2}^n a_k b_k \\ &= a_1 B_1 + \sum_{k=2}^n a_k (B_k - B_{k-1}) \\ &= a_1 B_1 + \sum_{k=2}^n a_k B_k - \sum_{k=2}^n a_k B_{k-1} \\ &= \sum_{k=1}^{n-1} a_k B_k + a_n B_n - \sum_{k=1}^{n-1} a_{k+1} B_k \\ &= \sum_{k=1}^{n-1} (a_k - a_{k+1}) B_k + a_n B_n \end{aligned}$$