Evaluating Infinite Series Involving Harmonic Numbers by Integration

Abstract

Eight infinite series involving harmonic-like numbers are coherently and systematically reviewed. They are evaluated in closed form exclusively by integration together with calculus and complex analysis. In particular, a mysterious series W is introduced and shown to be expressible in terms of the trilogarithm function. Several remarkable integral values and difficult infinite series identities are shown as consequences.

Keywords

• Euler sum
• harmonic number
• trilogarithm
• Catalan’s constant
• Riemann zeta function