proof of Abel’s lemma (by induction)

Proof. The proof is by inductionMathworldPlanetmath. However, let us first recall that sum on the right side is a piece-wise defined function of the upper limitMathworldPlanetmath N-1. In other words, if the upper limit is below the lower limit 0, the sum is identically set to zero. Otherwise, it is an ordinary sum. We therefore need to manually check the first two cases. For the trivial case N=0, both sides equal to a0b0. Also, for N=1 (when the sum is a normal sum), it is easy to verify that both sides simplify to a0b0+a1b1. Then, for the induction step, suppose that the claim holds for some N1. For N+1, we then have

i=0N+1aibi = i=0Naibi+aN+1bN+1
= i=0N-1Ai(bi-bi+1)+ANbN+aN+1bN+1
= i=0NAi(bi-bi+1)-AN(bN-bN+1)+ANbN+aN+1bN+1.

Since -AN(bN-bN+1)+ANbN+aN+1bN+1=AN+1bN+1, the claim follows. .

Title proof of Abel’s lemma (by induction)
Canonical name ProofOfAbelsLemmabyInduction
Date of creation 2013-03-22 13:38:04
Last modified on 2013-03-22 13:38:04
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Proof
Classification msc 40A05