In this paper, we present various asymptotic series for the harmonic number $$H_n=\sum _{k=1}^{n}\frac{1}{k}$$ Hn=∑k=1n1k . For example, we give a pair of recurrence relations for determining the constants $$a_\ell $$ aℓ and $$b_\ell $$ bℓ such that $$\begin{aligned} H_n\sim \frac{1}{2}\ln \left( 2m+\frac{1}{3}\right) + \gamma +\sum _{\ell =1}^{\infty }\frac{a_\ell }{(2m+b_\ell )^ {2\ell }}\quad \text {as}\ n\rightarrow \infty , \end{aligned}$$ Hn∼12ln2m+13+γ+∑ℓ=1∞aℓ(2m+bℓ)2ℓasn→∞, where $$m =\tfrac{1}{2}n(n+1)$$ m=12n(n+1) ($$n\in {\mathbb {N}}:=\{1,2,\ldots \}$$ n∈N:={1,2,…} ) is the n-th triangular number and $$\gamma $$ γ is the Euler–Mascheroni constant.