On Qi’s Normalized Remainder of Maclaurin Power Series Expansion of Logarithm of Secant Function
In the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ln</mi><mi>sec<...
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2024-12-01
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| author | Hong-Chao Zhang Bai-Ni Guo Wei-Shih Du |
| author_facet | Hong-Chao Zhang Bai-Ni Guo Wei-Shih Du |
| author_sort | Hong-Chao Zhang |
| collection | DOAJ |
| description | In the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ln</mi><mi>sec</mi><mi>x</mi><mo>=</mo><mo>−</mo><mi>ln</mi><mi>cos</mi><mi>x</mi></mrow></semantics></math></inline-formula>; in view of a monotonicity rule for the ratio of two Maclaurin power series and by virtue of the logarithmic convexity of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msup><mn>2</mn><mi>x</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>ζ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></semantics></math></inline-formula>, they prove the logarithmic convexity of Qi’s normalized remainder; with the aid of a monotonicity rule for the ratio of two Maclaurin power series, the authors present the monotonic property of the ratio between two Qi’s normalized remainders. |
| format | Article |
| id | doaj-art-1cd686565aeb4668879db72c4b8a31d7 |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-1cd686565aeb4668879db72c4b8a31d72025-08-20T02:00:51ZengMDPI AGAxioms2075-16802024-12-01131286010.3390/axioms13120860On Qi’s Normalized Remainder of Maclaurin Power Series Expansion of Logarithm of Secant FunctionHong-Chao Zhang0Bai-Ni Guo1Wei-Shih Du2School of Mathematics and Physics, Hulunbuir University, Hulunbuir 021008, ChinaSchool of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, ChinaDepartment of Mathematics, National Kaohsiung Normal University, Kaohsiung 824004, TaiwanIn the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ln</mi><mi>sec</mi><mi>x</mi><mo>=</mo><mo>−</mo><mi>ln</mi><mi>cos</mi><mi>x</mi></mrow></semantics></math></inline-formula>; in view of a monotonicity rule for the ratio of two Maclaurin power series and by virtue of the logarithmic convexity of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msup><mn>2</mn><mi>x</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>ζ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></semantics></math></inline-formula>, they prove the logarithmic convexity of Qi’s normalized remainder; with the aid of a monotonicity rule for the ratio of two Maclaurin power series, the authors present the monotonic property of the ratio between two Qi’s normalized remainders.https://www.mdpi.com/2075-1680/13/12/860Bernoulli numberDirichlet eta functionMaclaurin power series expansionRiemann zeta functionStirling number of the second kindQi’s normalized remainder |
| spellingShingle | Hong-Chao Zhang Bai-Ni Guo Wei-Shih Du On Qi’s Normalized Remainder of Maclaurin Power Series Expansion of Logarithm of Secant Function Axioms Bernoulli number Dirichlet eta function Maclaurin power series expansion Riemann zeta function Stirling number of the second kind Qi’s normalized remainder |
| title | On Qi’s Normalized Remainder of Maclaurin Power Series Expansion of Logarithm of Secant Function |
| title_full | On Qi’s Normalized Remainder of Maclaurin Power Series Expansion of Logarithm of Secant Function |
| title_fullStr | On Qi’s Normalized Remainder of Maclaurin Power Series Expansion of Logarithm of Secant Function |
| title_full_unstemmed | On Qi’s Normalized Remainder of Maclaurin Power Series Expansion of Logarithm of Secant Function |
| title_short | On Qi’s Normalized Remainder of Maclaurin Power Series Expansion of Logarithm of Secant Function |
| title_sort | on qi s normalized remainder of maclaurin power series expansion of logarithm of secant function |
| topic | Bernoulli number Dirichlet eta function Maclaurin power series expansion Riemann zeta function Stirling number of the second kind Qi’s normalized remainder |
| url | https://www.mdpi.com/2075-1680/13/12/860 |
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