This is an amazing identity by Abel.
Let 
![\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).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3Dm%7D%5En+f_k%28g_%7Bk%2B1%7D-g_k%29%3D%5Bf_%7Bn%2B1%7Dg_%7Bn%2B1%7D-f_mg_m%5D-%5Csum_%7Bk%3Dm%7D%5En+g_%7Bk%2B1%7D%28f_%7Bk%2B1%7D-f_k%29.&bg=ffffff&fg=333333&s=0 "\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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