proof of Abel lemma (by expansion)

1 Abel lemma

i=0naibi=i=0n-1Ai(bi-bi+1)+Anbn, (1)

where, Ai=k=0iak. Sequences {ai}, {bi}, i=0,,n, are real or complex one.

2 Proof

We consider the expansion of the sum


on two different forms, namely:

  1. 1.

    On the short way.

    i=0nAi(bi-bi+1)=i=0n-1Ai(bi-bi+1)+Anbn-Anbn+1. (2)
  2. 2.

    On the long way.

A0b0+i=1n(Ai-1+ai)bi-i=1nAi-1bi-Anbn+1=i=0naibi-Anbn+1, (3)

where a simplification has been performed. Notice that A0=a0. By equating (2), (3), the last two terms cancel, 11Without loss of generality, bn+1 may be assumed finite. Indeed we don’t need bn+1, but the proof is a couple lines larger. It is left as an exercise. and then, (1) follows.

Title proof of Abel lemma (by expansion)
Canonical name ProofOfAbelLemmabyExpansion
Date of creation 2013-03-22 17:28:14
Last modified on 2013-03-22 17:28:14
Owner perucho (2192)
Last modified by perucho (2192)
Numerical id 7
Author perucho (2192)
Entry type Proof
Classification msc 40A05