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).

 

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