Abel’s lemma

Theorem 1 Let {ai}i=0N and {bi}i=0N be sequences of real (or complex) numbers with N0. For n=0,,N, let An be the partial sum An=i=0nai. Then


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 {ai}i=MN and {bi}i=MN be sequences of real (or complex) numbers with 0MN. For n=M,,N, let An be the partial sum An=i=Mnai. Then


Proof. By defining a0==aM-1=b0==bM-1=0, we can apply Theorem 1 to the sequences {ai}i=0N and {bi}i=0N.


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 partsPlanetmathPlanetmath
Synonym Abel’s partial summation
Synonym Abel’s identity
Synonym Abel’s transformation
Related topic PartialSummation