Summation by Parts

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

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

Alternative formula
If we write $\displaystyle G_n = \sum_{k = 1}^n g_k$, we have for $m \geq 2$,

When $m = 1$, the same derivation gives


 * $\displaystyle \sum_{k = 1}^n f_k g_k = f_n G_n - \sum_{k = 1}^{n - 1} G_k\left( f_{k+1} - f_k \right)$

This is sometimes called the Abel transformation or Abel's lemma.