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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/13/12/860 |
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| Summary: | 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. |
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| ISSN: | 2075-1680 |