## Summation by parts / Abel’s Lemma

This is an amazing identity by Abel.

Let $\{f_k\}$ and $\{g_k\}$ be two sequences. Then,
$\displaystyle \sum_{k=m}^n f_k(g_{k+1}-g_k)=[f_{n+1}g_{n+1}-f_mg_m]-\sum_{k=m}^n g_{k+1}(f_{k+1}-f_k).$