Asymptotic inequalities for alternating harmonics

Abstract

For n ∈ ℕ the nth alternating harmonic number Hn∗ :=∑ k=1n(−1)k−1 1 k is given in the form Hn∗ =ln 2 + (−1)n+1 4⌊n+1 2 ⌋ +∑i=1q−1 (4i − 1)B 2i (2i)(2⌊n+1 2 ⌋)2i + rq(n), where q ∈ ℕ is a parameter controlling the magnitude of the error term rq(n) estimated as 0 < (−1)q+1r q(n) < |B2q| 2q ⋅⌊n+1 2 ⌋2q < 2 exp( 1 24q) 1 − 2 ⋅ 4−qπ q q eπ⌊n+1 2 ⌋2q.

Keywords

• approximation
• double inequality
• estimate
• alternating harmonic number