Monotone and fast computation of Euler's constant.

Area of Science:

• Mathematical Analysis
• Number Theory
• Probability Theory

Background:

• The Euler-Mascheroni constant (γ) is a fundamental mathematical constant.
• Efficient computation and representation of γ are of ongoing interest.
• Existing methods for approximating γ may lack speed or computational simplicity.

Purpose of the Study:

• To construct novel sequences that converge to the Euler-Mascheroni constant (γ).
• To achieve a geometric convergence rate of 1/2 for these sequences.
• To derive a new infinite product representation for γ.

Main Methods:

• Construction of increasing and decreasing finite sum sequences.
• Demonstration of complete monotonicity-type properties for the sequences.
• Utilizing a probabilistic approach involving the gamma process and differentiation formulas.

Main Results:

• Two sequences were constructed, converging increasingly and decreasingly to γ at a geometric rate of 1/2.
• The sequences exhibit desirable computational properties and monotonicity.
• An infinite product representation for γ was derived, converging monotonically and rapidly.

Conclusions:

• The newly constructed sequences provide an efficient and fast method for approximating the Euler-Mascheroni constant.
• The derived infinite product representation offers a novel approach to representing γ.
• The probabilistic method provides a powerful tool for analyzing mathematical constants.