Summation by parts

In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. The summation by parts formula is sometimes called Abel's lemma or Abel transformation.

Suppose ${\displaystyle \{f_{k}\}}$ and ${\displaystyle \{g_{k}\}}$ are two sequences. Then,

Using the forward difference operator ${\displaystyle \Delta }$, it can be stated more succinctly as

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