summation by parts


The following corollaries apply Abel’s lemma to allow estimation of certain bounded sums:

Corollary 1

(Summation by partsPlanetmathPlanetmath)
Let {ai},{bi} be sequencesPlanetmathPlanetmath of complex numbers. Suppose the partial sums of the ai are bounded in magnitude by h, that 0|bi-bi+1| converges, and that limibi=0. Then 0aibi converges, and

|0aibi|h0|bi-bi+1|

Proof. By Abel’s lemma,

i=0Naibi=i=0N-1Ai(bi-bi+1)+ANbN

so that

|i=0Naibi| =|i=0N-1Ai(bi-bi+1)+ANbN|i=0N-1|Ai(bi-bi+1)|+|ANbN|
hi=0N-1|bi-bi+1|+h|bN|

The condition that the bi0 is easily seen to imply that the sequence |i=0Naibi| is Cauchy hence convergent, so that

|i=0aibi|hi=0|bi-bi+1|

since bN0.

Corollary 2

(Summation by parts for real sequences)
Let {ai} be a sequence of complex numbers. Suppose the partial sums are bounded in magnitude by h. Let {bi} be a sequence of decreasing positive real numbers such that limibi=0. Then 1aibi converges, and |1aibi|hb1.

Proof. This follows immediately from the above, since |bi-bi+1|=bi-bi+1.

Title summation by parts
Canonical name SummationByParts
Date of creation 2013-03-22 16:28:10
Last modified on 2013-03-22 16:28:10
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 8
Author rm50 (10146)
Entry type Theorem
Classification msc 40A05
Classification msc 40D05
Synonym partial summation