Abstract
The problem of completely describing the approximation of the number \(e\) by the elements of the sequence \((1+1/m)^m\), \(m\in\mathbb{N}\), is considered. To this end, the function \(f(z)=\exp\{(1/z)\ln(1+z)-1\}\), which is analytic in the complex plane with a cut along the half-line \((-\infty,-1]\) of the real line, is studied in detail. We prove that the power series \(1+\sum^{\infty}_{n=1}(-1)^n a_n z^n\), where all \(a_n\) are positive, which represents this function on the unit disk, envelops it in the open right half-plane. This gives a series of double inequalities for the deviation \(e-(1+x)^{1/x}\) on the positive half-line, which are asymptotically sharp as \(x\to 0\). Integral representations of the function \(f(z)\) and of the coefficients \(a_n\) are obtained. They play an important role in the study. A two-term asymptotics of the coeffients \(a_n\) as \(n\to \infty\) is found. We show that the coefficients form a logarithmically convex completely monotone sequence. We also obtain integral expressions for the derivatives of all orders of the function \(f(z)\). It turns out that \(f(x)\) is completely monotone on the half-line \(x>-1\). Applications and development of the results are discussed.
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Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 374–391 https://doi.org/10.4213/mzm13716.
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Kostin, A.B., Sherstyukov, V.B. Enveloping of the Values of an Analytic Function Related to the Number \(e\). Math Notes 113, 368–383 (2023). https://doi.org/10.1134/S0001434623030069
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DOI: https://doi.org/10.1134/S0001434623030069