We consider a one-parameter family of number series involving the generalized harmonic series and study asymptotic properties of the remainders. Using \( R\left(N,p\right)\equiv \sum \limits_{n=N}^{\infty }1/{n}^p \) as an example, we describe the typical obtained results: we obtain the integral representation, find the complete asymptotic expansion with respect to the parameter 2N − 1 as N →∞, and prove that R(N, p) is enveloped by its asymptotic series. The possibilities of the proposed approach are demonstrated by the problem of exact two-sided estimates for the central binomial coefficient.
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Translated from Problemy Matematicheskogo Analiza 107, 2020, pp. 39-58.
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Kostin, A.B., Sherstyukov, V.B. Asymptotic Behavior of Remainders of Special Number Series. J Math Sci 251, 814–838 (2020). https://doi.org/10.1007/s10958-020-05131-2
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DOI: https://doi.org/10.1007/s10958-020-05131-2