Abel’s lemma

Theorem 1 Let ${\{a_i\}}_{i=0}^N$ and ${\{b_i\}}_{i=0}^N$ be sequences of real (or complex) numbers with $N\ge 0$. For $n=0,\mathrm{\dots },N$, let $A_n$ be the partial sum $A_n={\sum }_{i=0}^n{a}_i$. Then

$$\sum _{i=0}^N{a}_i{b}_i=\sum _{i=0}^{N-1}{A}_i(b_i-b_{i+1})+A_N{b}_N.$$

In the trivial case, when $N=0$, then sum on the right hand side should be interpreted as identically zero. In other words, if the upper limit is below the lower limit, there is no summation.

An inductive proof can be found here (http://planetmath.org/ProofOfAbelsLemmaByInduction). The result can be found in [1] (Exercise 3.3.5).

If the sequences are indexed from $M$ to $N$, we have the following variant:

Corollary Let ${\{a_i\}}_{i=M}^N$ and ${\{b_i\}}_{i=M}^N$ be sequences of real (or complex) numbers with $0\le M\le N$. For $n=M,\mathrm{\dots },N$, let $A_n$ be the partial sum $A_n={\sum }_{i=M}^n{a}_i$. Then

$$\sum _{i=M}^N{a}_i{b}_i=\sum _{i=M}^{N-1}{A}_i(b_i-b_{i+1})+A_N{b}_N.$$

Proof. By defining $a_0=\mathrm{\dots }=a_{M-1}=b_0=\mathrm{\dots }=b_{M-1}=0$, we can apply Theorem 1 to the sequences ${\{a_i\}}_{i=0}^N$ and ${\{b_i\}}_{i=0}^N$. $\mathrm{\square }$

Title	Abel’s lemma
Canonical name	AbelsLemma
Date of creation	2013-03-22 13:19:49
Last modified on	2013-03-22 13:19:49
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	14
Author	mathcam (2727)
Entry type	Theorem
Classification	msc 40A05
Synonym	summation by parts
Synonym	Abel’s partial summation
Synonym	Abel’s identity
Synonym	Abel’s transformation
Related topic	PartialSummation