proof of Abel lemma (by expansion)

1 Abel lemma

 $\sum_{i=0}^{n}a_{i}b_{i}=\sum_{i=0}^{n-1}A_{i}(b_{i}-b_{i+1})+A_{n}b_{n},$ (1)

where, $A_{i}=\sum_{k=0}^{i}a_{k}$. Sequences $\{a_{i}\}$, $\{b_{i}\}$, $i=0,\dots,n$, are real or complex one.

2 Proof

We consider the expansion of the sum

 $\sum_{i=0}^{n}A_{i}(b_{i}-b_{i+1})$

on two different forms, namely:

1. 1.

On the short way.

 $\sum_{i=0}^{n}A_{i}(b_{i}-b_{i+1})=\sum_{i=0}^{n-1}A_{i}(b_{i}-b_{i+1})+A_{n}b% _{n}-A_{n}b_{n+1}.$ (2)
2. 2.

On the long way.

 $\sum_{i=0}^{n}A_{i}(b_{i}-b_{i+1})=\sum_{i=0}^{n}A_{i}b_{i}-\sum_{i=0}^{n}A_{i% }b_{i+1}=\sum_{i=0}^{n}A_{i}b_{i}-\sum_{i=1}^{n+1}A_{i-1}b_{i}=$
 $A_{0}b_{0}+\sum_{i=1}^{n}(A_{i-1}+a_{i})b_{i}-\sum_{i=1}^{n}A_{i-1}b_{i}-A_{n}% b_{n+1}=\sum_{i=0}^{n}a_{i}b_{i}-A_{n}b_{n+1},$ (3)

where a simplification has been performed. Notice that $A_{0}=a_{0}$. By equating (2), (3), the last two terms cancel, 11Without loss of generality, $b_{n+1}$ may be assumed finite. Indeed we don’t need $b_{n+1}$, but the proof is a couple lines larger. It is left as an exercise. and then, (1) follows. $\Box$

Title proof of Abel lemma (by expansion) ProofOfAbelLemmabyExpansion 2013-03-22 17:28:14 2013-03-22 17:28:14 perucho (2192) perucho (2192) 7 perucho (2192) Proof msc 40A05