summation by parts
The following corollaries apply Abel’s lemma to allow estimation of certain bounded sums:
Proof. By Abel’s lemma,
The condition that the is easily seen to imply that the sequence is Cauchy hence convergent, so that
(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|
|Date of creation||2013-03-22 16:28:10|
|Last modified on||2013-03-22 16:28:10|
|Last modified by||rm50 (10146)|