Summation by parts
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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 <math>\{a_k\}<math> and <math>\{b_k\}<math> are two sequences. Then,
- <math>\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)<math>
Using the difference operator, it can be stated as more succinctly as
- <math>\sum a_k\Delta b_k = a_{k+1}b_{k+1} - \sum b_{k+1}\Delta a_k,<math>
as an analogue to the integration by parts formula,
- <math>\int u\,dv = uv - \int v\,du.<math>
The summation by parts formula is sometimes called Abel's lemma.
External link
- "Abel's lemma" article at PlanetMath.org (http://planetmath.org/encyclopedia/AbelsLemma.html)