Summation by Parts

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

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

Proof
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