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. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826.

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

Or, equivalently, using backward difference operator ${\displaystyle \nabla }$:

Both can be used to obtain a more symmetric version:

Summation by parts is an analogue to integration by parts:

or to Abel's summation formula:

An alternative statement is