Summation by parts for discrete variables is the equivalent of integration by parts for continuous variables.
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 $ \lbrace f_{k} \rbrace $ and $ \lbrace g_{k} \rbrace $ are two sequences. Then,
$$ \sum _ {k=m} ^ {n} f _ {k} (g _ {k+1} - g _ {k} )= \left( f _ {n} g _ {n+1} - f _ {m} g _ {m} \right) - \sum _ {k=m+1} ^ {n} g _ {k} (f _ {k} - f _ {k-1} ) \text . $$
Using the forward difference operator $ \Delta $, it can be stated more succinctly as
$$ \sum _ {k=m} ^ {n} f _ {k} \Delta g _ {k} = \left( f _ {n} g _ {n+1} - f _ {m} g _ {m} \right) - \sum _ {k=m} ^ {n-1} g _ {k+1} \Delta f _ {k} \text , $$
Summation by parts is an analogue to integration by parts:
$$ \int f \ dg = f g - \int g \ df \text , $$
or to Abel's summation formula:
$$ \sum _ {k=m+1} ^ {n} f(k) (g _ {k} - g _ {k-1} ) = \left( f(n) g _ {n} - f(m) g _ {m} \right) - \int _ {m} ^ {n} g _ { \lfloor t \rfloor } f'(t) dt \text . $$
An alternative statement is
$$ f _ {n} g _ {n} - f _ {m} g _ {m} = \sum _ {k=m} ^ {n-1} f _ {k} \Delta g _ {k} + \sum _ {k=m} ^ {n-1} g _ {k} \Delta f _ {k} + \sum _ {k=m} ^ {n-1} \Delta f _ {k} \Delta g _ {k} $$
which is analogous to the integration by parts formula for semimartingales.
Although applications almost always deal with convergence of sequences, the statement is purely algebraic and will work in any field. It will also work when one sequence is in a vector space, and the other is in the relevant field of scalars.
