Summation by parts

Summation by parts rewrites a finite sum of products as boundary terms plus a sum involving finite differences. It is the discrete analogue of integration by parts.

Discrete product rule: A rule expressing the difference of a product in terms of differences of its factors. Summation by parts rearranges this rule and sums it over an interval.
Abel's test: A convergence test for series combining bounded partial sums with a monotone sequence tending to zero. Its proof uses partial summation to bound weighted sums.
Sequence: An ordered list of values indexed by integers or another specified set. Summation by parts operates on products of indexed sequences.
Finite summation-by-parts formula: An identity expressing a finite sum of paired sequences through endpoint products and successive differences. This is the standard formula underlying the concept.
Finite difference: An operation that measures changes in a function across discrete input values. The identity replaces one factor with its finite differences.
Dirichlet's test: A test for convergence of a series when one sequence has bounded partial sums and another decreases to zero. Summation by parts turns these conditions into a convergence bound.
Partial sum: The sum of the initial terms of a sequence up to a specified index. Accumulating one factor into partial sums gives a common form of the identity.
Abel's lemma: A finite-sum identity relating a weighted sum to partial sums and differences of the weights. It presents summation by parts in a frequently used sequence notation.
Telescoping sum: A sum whose successive terms cancel, leaving only boundary contributions. The boundary terms arise when the discrete product rule is summed.
Euler–Maclaurin formula: A formula relating a sum to an integral through endpoint corrections and derivatives. Repeated summation by parts helps derive its correction terms.

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