The fractal characteristics of the Catalan sequence

The fractal characteristics of the Catalan sequence and its reciprocal version have been systematically investigated. In particular, the box dimension and the Hausdorff dimension of the reciprocal sequence are rigorously calculated. It has been proved that both the box and the Hausdorff dimensions of the reciprocal sequence are equal to zero, which reflects the ultra-fast convergence rates of the sequence toward zero. Furthermore, the convergence of the series formed by the reciprocal Catalan sequence is verified based on the fractal dimension criterion and the double monotonicity of the sequence. Numerical experiments have been conducted to validate the theoretical results. The results obtained provide a theoretical framework for the fractal analysis of classical combinatorial sequences and demonstrate the intrinsic relationship between the fractal dimension of point sequences and their convergence rates.