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- Title
ESTIMATING THE REMAINDER OF AN ALTERNATING p–SERIES USING HYPERGEOMETRIC FUNCTIONS.
- Authors
ECHI, OTHMAN; KHALFALLAH, ADEL; KROUMI, DHAKER
- Abstract
In this paper, using hypergeometric functions, we provide sharp estimates of the remainder of the alternating p-series, ∑n≥1/(−1)n−1 np, where p ≥ 2 is an integer. We show that the largest ρ and the largest σ such that the inequalities 1/2(n+1)p−ρ≤| ∑k=n+1∞ (−1)k−1/kp|≤ 1/2np +σ, hold for any integer n ≥ 1 are ρ (p) = 2p+1− 1/1−(1−2¹−p)ζ (p) and σ (p) = 1/1−(1−21−p)ζ (p) −2, where ζ (p) = ∑k=1∞ 1/kp, the Riemann zeta function.
- Subjects
HYPERGEOMETRIC functions; MATHEMATICAL programming; INTEGERS; RATIONAL numbers; RIEMANN integral
- Publication
Journal of Mathematical Inequalities, 2023, Vol 17, Issue 2, p569
- ISSN
1846-579X
- Publication type
Article
- DOI
10.7153/jmi-2023-17-36