Asymptotic Behavior of Remainders of Special Number Series
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Journal of Mathematical Sciences, Vol. 251, No. 6, December, 2020
ASYMPTOTIC BEHAVIOR OF REMAINDERS OF SPECIAL NUMBER SERIES A. B. Kostin ∗ National Research Nuclear University MEPhI 31, Kashirskoe shosse, Moscow 115409, Russia [email protected]
V. B. Sherstyukov National Research Nuclear University MEPhI 31, Kashirskoe shosse, Moscow 115409, Russia [email protected]
UDC 517.984.5, 517.584
We consider a one-parameter family of number series involving the generalized harmonic ∞ 1/np series and study asymptotic properties of the remainders. Using R(N, p) ≡ n=N
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. Bibliography: 15 titles.
1
Introduction
The number series ∞ 1 , np
n=1
∞ (−1)n , np
n=1
∞ m=0
1 , (2m + 1)p
∞ m=0
(−1)m , (2m + 1)p
with parameter p ∈ C are well known mathematical objects. The first series in the half-plane Re p > 1 presents the famous Riemann zeta function, the second series is the value at z = −1 of the polylogarithm ∞ zn , Lip (z) ≡ np n=1
the third and fourth ones are directly connected with the Favard constants in approximation ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 107, 2020, pp. 39-58. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2516-0814
814
theory. The last series or, more exactly, the quantity ∞
β(p) ≡
m=0
(−1)m (2m + 1)p
is called the Dirichlet beta function (also known as the Catalan beta function), and the number G ≡ β(2) =
∞ m=0
(−1)m (2m + 1)2
is known as the Catalan constant. We are interested in asymptotic properties of the remainders of such series with integer parameter p. To simplify the exposition, we set p = s + 1 and for N ∈ N denote ∞
R(N, s) ≡
1 ns+1
n=N
R∗ (N, s) ≡
,
s ∈ N,
∞ (−1)n , ns+1
(1.1)
s ∈ N ∪ {0} ,
(1.2)
n=N
r(N, s) ≡
∞ m=N
r∗ (N, s) ≡
1 , (2m + 1)s+1
∞ m=N
(−1)m , (2m + 1)s+1
s ∈ N,
s ∈ N ∪ {0} .
(1.3)
(1.4)
Now, we describe the structure of the paper and formulate the main results. Section 2 contains technical lemmas which are used in the proof of the following assertion. Theorem 1.1. Assume that s ∈ N∪{0}, y ∈ C[0, +∞)∩C (s+1) (0, +∞), and f (t) ≡ y (s+1) (t), t > 0. If y(t) and f (t) satisfy the conditions (1) tk y (k) (t) → 0 as t → +∞ (for every k = 0, . . . , s), +∞ τ s f (τ )dτ converges, (2) the integral 0
then the following representation holds: (−1)s+1 y(t) = s!
+∞ (τ − t)s f (τ )dτ,
t 0,
t
in particular, (−1)s+1 y(0) = s!
+∞ τ s f (τ )dτ. 0
Theorem 1.1 is used in Section 2 to obtain the integral representations of the remainders (1.1)–(1.4). 815
Theorem 1.2. For any N ∈ N the remainders (1.1)–(1.4) admit the integral representations R(N, s) ≡
∞
1 ns+1
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