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 rule states:

Suppose ${a k}$ and ${b k}$ are two sequences. Then,

$\sum_{k=m}^n a_k(b_{k+1}-b_k) = \left[a_{n+1}b_{n+1} - a_mb_m\right] - \sum_{k=m}^n b_{k+1}(a_{k+1}-a_k)$

Using the difference operator, it can be stated as more succinctly as

$\sum a_k\Delta b_k = a_{k+1}b_{k+1} - \sum b_{k+1}\Delta a_k,$

as an analogue to the integration by parts formula,

$\int u\,dv = uv - \int v\,du.$

The summation by parts formula is sometimes called Abel's lemma.

External link

"Abel's lemma" article at PlanetMath.org

Summation by parts

External link

See also: Summation by parts, Difference operator, Integration by parts, Mathematics, Sequence, Summation