A series representation for the Catalan constant

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Published/Copyright: August 29, 2025

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From the journal Georgian Mathematical Journal

Abstract

We prove the following series representation for the Catalan constant G:

G = 3 10 - 48 ⁢ ∑ n = 0 ∞ 4 ⁢ n 2 + 8 ⁢ n - 9 ( 4 ⁢ n 2 - 4 ⁢ n + 5 ) ⁢ ( 4 ⁢ n 2 + 20 ⁢ n + 29 ) ⁢ S n 3 ,

where

S n = ∑ k = 0 n ( - 1 ) k 2 ⁢ k + 1

is the nth partial sum of the classical Leibniz series for π 4 .

MSC 2020: 11A67; 26A99

I thank the referee for the kind and inspiring comments.

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Received: 2025-06-27

Revised: 2025-07-23

Accepted: 2025-07-28

Published Online: 2025-08-29

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https://doi.org/10.1515/gmj-2025-2070

Keywords for this article

Catalan constant; Leibniz series; series representation