Evaluation of harmonic number series involving the binomial coefficient $C(3n,n)$ in the denominator by integration

TL;DR

This paper evaluates two classes of infinite series involving harmonic numbers and the binomial coefficient C(3n,n) using integral methods, deriving closed-form expressions and special case identities.

Contribution

It introduces novel integral techniques to evaluate complex harmonic series involving C(3n,n), providing new closed-form formulas and identities.

Findings

Closed-form expressions for series involving harmonic numbers and C(3n,n)

Identification of remarkable integral values and identities

Simplification of difficult series into integral representations

Abstract

Two classes of infinite series involving harmonic numbers and the binomial coefficient $C(3n,n)$ are evaluated in closed form using integrals. Several remarkable integral values and difficult series identities are stated as special cases of the main results.