Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 41, 24 June 2024
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A method to test the uniform convergence of function series
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Abstract
The series refer to performing infinite addition operations on infinite numbers or functions in a certain order. It is hard to find out whether the positive function series converges uniformly in many cases. In this article, a new method that replacing the sum of function terms series with improper integral will be introduced, which is designed to solve problems that cannot be solved by classical Weierstrass M-test. The Cauchy uniform convergence test will serve as the basis for the entire proof process because it can lead the focus point from the whole sum to the partial sum of the function series, where its value can be easier substituting by the value of the improper integral. After using basic knowledge of the improper integral, the uniform convergence can finally be known. By using this method, testing the uniform convergence of the irregular function series even estimating its value can be possible accomplished.
Keywords
Uniform convergence of function series, Improper integral, Cauchy’s convergence test, Weierstrass M-test, Mathematical analysis
References
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Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
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- Volume Title
- Proceedings of Machine Learning: Integrating machine learning techniques to advance network security - CONFMPCS 2024
- ISBN (Print)
- 978-1-83558-493-4
- ISBN (Online)
- 978-1-83558-494-1
- Published Date
- 24 June 2024
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/41/20240105
- Copyright
- 24 June 2024
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited