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A series of sequences convergent to Euler's constant
Li-Jiang Jia1, Bin Ge2, Li-Li Liu2
11School of Economics and Management, Harbin Engineering University, Harbin, P.R. China.
Journal of Inequalities and Applications
|August 24, 2018
Summary
Researchers developed a faster sequence converging to Euler's constant using continued fractions. This new method surpasses existing sequences in speed and efficiency.
Area of Science:
- Number Theory
- Mathematical Analysis
Background:
- Euler's constant (gamma) is a fundamental mathematical constant.
- Efficiently approximating Euler's constant is crucial for various mathematical computations.
- Existing sequences for approximating Euler's constant have limitations in convergence speed.
Purpose of the Study:
- To introduce a novel sequence for approximating Euler's constant.
- To establish the superiority of the new sequence over previously known sequences.
- To enhance the efficiency of calculating Euler's constant.
Main Methods:
- The study utilizes the theory of continued fractions.
- A new sequence is constructed based on continued fraction expansions.
- Convergence rates are analyzed and compared with existing methods.
Main Results:
- The proposed sequence demonstrates a quicker convergence rate to Euler's constant.
- The new sequence is shown to be more efficient than sequences developed by DeTemple, Mortici, Vernescu, and Lu.
- Mathematical proofs confirm the superior performance of the new convergent sequence.
Conclusions:
- The developed continued fraction-based sequence offers a significant improvement for approximating Euler's constant.
- This new method provides a faster and more efficient tool for mathematicians and scientists.
- The findings contribute to the ongoing research in the numerical approximation of mathematical constants.
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