A new sequence related to the Euler-Mascheroni constant

Area of Science:

• Number Theory
• Mathematical Analysis

Background:

• The Euler-Mascheroni constant (γ) is a fundamental mathematical constant.
• Efficient computation of mathematical constants is crucial in various scientific fields.
• Existing sequences for approximating γ have limitations in convergence speed or complexity.

Purpose of the Study:

• To introduce a new sequence for approximating the Euler-Mascheroni constant.
• To enhance the speed of convergence compared to existing methods.
• To provide rigorous error bounds for the new approximation sequence.

Main Methods:

• Development of a novel sequence based on Padé-type approximation.
• Analysis of the convergence properties of the new sequence.
• Derivation of lower and upper bounds for the approximation error.

Main Results:

• A new sequence with a simple form and significantly faster convergence to γ is presented.
• The established bounds provide a measure of the approximation's accuracy.
• The proposed method demonstrates superior performance over traditional approaches.

Conclusions:

• The new sequence offers an efficient and accurate method for approximating the Euler-Mascheroni constant.
• This advancement has potential applications in numerical analysis and computational mathematics.
• The findings contribute to the ongoing research in the efficient computation of mathematical constants.