[Paper Review] Series representations of the remainders in the expansion for certain trigonometric and hyperbolic functions with applications

C.-P. Chen, R. B. Paris|arXiv (Cornell University)|Jan 13, 2016

Mathematical Inequalities and Applications20 references3 citations

TL;DR

This paper derives explicit series representations for the remainders in Taylor expansions of trigonometric and hyperbolic functions, particularly tan t, tanh t, and sec t, using integral and infinite series identities. The key contribution is tight, sharp inequalities for these functions with best-possible constants, established via connections to Bernoulli and Euler numbers and asymptotic analysis of remainder terms.

ABSTRACT

In this paper, we present series representations of the remainders in the expansions for certain trigonometric and hyperbolic functions. By using the obtained results, we establish some inequalities for trigonometric and hyperbolic functions.

Motivation & Objective

• To derive explicit series representations for the remainder terms in the Taylor expansions of tan t, tanh t, and sec t.
• To establish sharp inequalities for trigonometric and hyperbolic functions using the derived remainder representations.
• To determine the best possible constants in these inequalities, particularly for sec t and tan t.
• To extend previous results on remainders in expansions of sech t and coth t to new functions using similar techniques.
• To provide a rigorous foundation for inequalities involving ratios of gamma functions through remainder analysis.

Proposed method

• Utilizes the infinite series identity: tan t = ∑_{k=1}^∞ 8t / (π²(2k−1)² − 4t²), derived from known partial fraction expansions.
• Applies the geometric series expansion 1/(1−q) = ∑_{j=0}^{N−1} q^j + q^N/(1−q) to decompose the remainder term into polynomial and residual components.
• Employs known identities for sums over odd integers: ∑_{k=1}^∞ 1/(2k−1)^{2n} = (2^{2n}−1)π^{2n}|B_{2n}|/(2·(2n)!), linking to Bernoulli numbers.
• Uses the reflection and recurrence properties of the trigamma function ψ′(z) to prove the series identity for ∑_{k=N+2}^∞ 1/(4k²−1)².
• Applies bounds on Euler numbers |E_{2n}|/(2n)! using the inequality 4^{n+1}/π^{2n+1}(1+3^{-1-2n}) < |E_{2n}|/(2n)! < 4^{n+1}/π^{2n+1} to establish sharpness.
• Derives remainder representations for tanh t and sec t via analytic continuation and sign analysis of series terms.

Experimental results

Research questions

• RQ1 What are the exact series representations for the remainder terms in the Taylor expansions of tan t and tanh t?
• RQ2 How can the remainder terms be bounded to yield sharp inequalities for sec t and tan t?
• RQ3 What are the best possible constants in the inequalities involving the remainder of sec t’s expansion?
• RQ4 How do the derived remainder representations improve upon classical inequalities like Becker–Stark?
• RQ5 Can the remainder structure be generalized to other trigonometric and hyperbolic functions using special functions and number sequences?

Key findings

• The remainder in the expansion of tan t is represented as ϑ_N(t) = (2^{2N+3}t^{2N+1}/π^{2N}) × ∑_{k=1}^∞ 1/[(2k−1)^{2N}(π²(2k−1)²−4t²)], valid for |t| < π/2.
• For tan t, the inequality 1 + (32t²)/(π²(π²−4t²)) + (32t²(π⁴/96−1))/π⁴ < tan t/t < 1 + (32t²)/(π²(π²−4t²)) + (8t²(5−π²/2))/π⁴ holds for 0 < t < π/2, improving on Becker–Stark.
• The remainder in tanh t’s expansion is τ_N(t) = (−1)^N × (2^{2N+3}t^{2N+1}/π^{2N}) × ∑_{k=1}^∞ 1/[(2k−1)^{2N}(π²(2k−1)²+4t²)], valid for all real t.
• For sec t, the inequality |E_{2N}|/(2N)! < (sec x − s_N(x))/(x^{2N−1} tan x) < (2/π)^{2N−1} holds for 0 < x < π/2, with both constants optimal.
• The limit behavior lim_{x→0} (sec x − s_N(x))/(x^{2N−1} tan x) = |E_{2N}|/(2N)! and lim_{x→π/2} = (2/π)^{2N−1} confirms the sharpness of the bounds.
• The paper proves the identity ∑_{k=N+2}^∞ 1/(4k²−1)² = (1/8)ψ′(N+3/2) − (N+2)/(2(2N+3)²) via induction and trigamma function recurrence.