summation by parts
The following corollaries apply Abel’s lemma to allow estimation of certain bounded sums:
Corollary 1
(Summation by parts)
Let be sequences of complex numbers. Suppose the partial sums of the are bounded in magnitude by , that converges, and that . Then converges, and
Proof. By Abel’s lemma,
so that
The condition that the is easily seen to imply that the sequence is Cauchy hence convergent, so that
since .
Corollary 2
(Summation by parts for real sequences)
Let be a sequence of complex numbers. Suppose the partial sums are bounded in magnitude by . Let be a sequence of decreasing positive real numbers such that . Then converges, and .
Proof. This follows immediately from the above, since .
Title | summation by parts |
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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 |