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Telescoping Estimates for Smooth Series
- Source :
- Canadian Mathematical Bulletin. 58:188-195
- Publication Year :
- 2015
- Publisher :
- Canadian Mathematical Society, 2015.
-
Abstract
- We derive telescoping majorants and minorants for some classes of series and give applications of these results. P ∞ k=n k −s , s > 1. The present article is dedicated to the question which se- ries can be treated in a similar way. In the following we shall show that this is the case for a big class of series P ∞=n f(k). They have to satisfy only certain mild smoothness conditions on the function f. The proofs are based on the comparison of f(k) and Z k+1−c k−c f(x) dx, c 2 (0,1), that may be regarded as special cases of theorems from the theory of numer- ical integration. We will demonstrate the usefulness of this method by some applications. Among them there will be a generalization of the Stieltjes constants and an elementary proof of Stirling's formula.
- Subjects :
- Telescoping series
Pure mathematics
Class (set theory)
Smoothness (probability theory)
Series (mathematics)
Generalization
General Mathematics
010102 general mathematics
Mathematical analysis
Stieltjes constants
010103 numerical & computational mathematics
Function (mathematics)
01 natural sciences
Elementary proof
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 14964287 and 00084395
- Volume :
- 58
- Database :
- OpenAIRE
- Journal :
- Canadian Mathematical Bulletin
- Accession number :
- edsair.doi...........de67ef685a6cad8a77302dcd4aa4ae94
- Full Text :
- https://doi.org/10.4153/cmb-2014-037-5