Computationally Efficient Bounds for the Sum of Catalan Numbers

TL;DR

This paper derives computationally efficient bounds for the sum of Catalan numbers, demonstrating that the average of these bounds outperforms previous bounds significantly, with practical implications for combinatorial calculations.

Contribution

It introduces new tight lower and upper bounds for the sum of Catalan numbers and proves the average of these bounds is a superior upper bound.

Findings

Lower bound is tighter than the previous upper bound.

The average of the bounds outperforms the previous upper bound by a factor greater than 4.5.

The bounds are easily computable and practically useful.

Abstract

Easily computable lower and upper bounds are found for the sum of Catalan numbers. The lower bound is proven to be tighter than the upper bound, which previously was declared to be only an asymptotic. The average of these bounds is proven to be also an upper bound, and empirically it is shown that the average is superior to the previous upper bound by a factor greater than (9/2).