Encyclopedia > Summation by parts

  Article Content

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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Christiania

...     Contents Christiania Christiania can refer to: Christiania - the name of Oslo, from 1624 to 1925. The Free State of Christiania - a partially ...

 
 
 
This page was created in 39.8 ms