Summation by parts

In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation 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(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_kb_k - \sum b_k\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.