Summation by Parts

Theorem
Let $$\left \langle {f_n} \right \rangle$$ and $$\left \langle {g_n} \right \rangle$$ be two sequences.

Then:
 * $$\sum_{k=m}^n f_k (g_{k+1} - g_n) = \left [{f_{n+1} g_{n+1} - f_m g_m}\right] - \sum_{k=m}^n g_{k+1}(f_{k+1}- f_k)$$