Accurate double inequalities for generalized harmonic numbers

doi:10.1016/j.amc.2015.04.128

Applied Mathematics and Computation

Volume 265, 15 August 2015, Pages 557-567

https://doi.org/10.1016/j.amc.2015.04.128 Get rights and content

Abstract

For $n \in N$ and $p \in R$ the nth harmonic number of order p $H (n, p) : = \sum_{k = 1}^{n} \frac{1}{k^{p}}$ is expressed in the form $H (n, p) = {\tilde{H}}_{q} (m, n, p) + R_{q} (m, n, p)$ where $m, q \in N$ are parameters controlling the magnitude of the error term. The function ${\tilde{H}}_{q} (m, n, p)$ consists of $m + 2 q + 1$ simple summands and the remainder R_q(m, n, p) is estimated, for p ≥ 0, as $\begin{matrix} 0 \leq {(- 1)}^{q + 1} R_{q} (m, n, p) < \frac{1}{π (1 - 2 \cdot 4^{- q})} {(\frac{\frac{p}{2} + q - 1)}{π m})}^{2 q - 1} \cdot \frac{1}{m^{p}} . \end{matrix}$ Similar result is obtained also for p < 0 and for real zeta function ( $n = \infty,$ p > 1) as well.

Introduction

The infinite convergent p-series, noted as Riemann zeta function, is well known, at least from the numerical stand point. However, the finite p-series, known also as the (generalized) nth harmonic number of order p, was studied less intensively. Only a few of authors, Abalo [1] and Hansheng [5], for example, have estimated this series. Recently, Chlebus [3] published the paper approximating the sequence of partial sums $H (n, p) : = \sum_{k = 1}^{n} \frac{1}{k^{p}}, p \in R .$ Using the probability techniques, Chlebus [3, rel. (33), (25, (26))] derived the following estimates: $C_{1}^{-} (n, p) : = 1 + \frac{n^{1 - p} - 1}{1 - p} < H (n, p) < 1 + \frac{{(n + 1)}^{1 - p} - 1}{1 - p} = : C_{2}^{-} (n, p)$ for p < 0, $C_{1}^{+} (n, p) : = \frac{{(n + 1)}^{1 - p} - 1}{1 - p} < H (n, p) < 1 + \frac{n^{1 - p} - 1}{1 - p} = : C_{2}^{+} (n, p)$ for p > 0, p ≠ 1, $\ln (n + 1) < H (n, 1) < 1 + \ln (n) .$

Our aim is to present more accurate approximations to harmonic numbers of order p than those given above. Indeed, we will improve the estimates (2)–(4), see Figs. 4 and 5, using the Euler–Maclaurin summation formula, the subject of the next section (see [4]).

Section snippets

Euler–Maclaurin summation formula

The sequence of Bernoulli polynomials B_n(x) is unambiguously defined inductively as $\begin{matrix} (i) & B_{0} (x) = 1, for x \in R \\ (i i) & B_{n}^{'} (x) = n B_{n - 1} (x), for n \in N and x \in R \\ (i i i) & \int_{0}^{1} B_{n} (x) d x = 0, for n \in N . \end{matrix}$ This way, for example, we obtain $B_{1} (x) \equiv x - 1 / 2,$ $B_{2} (x) \equiv x^{2} - x + 1 / 6,$ $B_{3} (x) \equiv x^{3} - 3 x^{2} / 2 + x / 2$ and $B_{4} (x) \equiv x^{4} - 2 x^{3} + x^{2} - 1 / 30$ .

The sequences of weighted Bernoulli polynomials V_n(x) and weighted Bernoulli 1-periodic functions W_n(x) are defined inductively as follows: $\begin{matrix} (i) & V_{0} (x) = 1, for x \in R, \\ (ii) & V_{n}^{'} (x) = V_{n - 1} (x), for n \in N and x \in R, \\ (iii) & V_{n} (0) : = V_{n} (1), for n \in N ∖ {1}, \end{matrix}$ and $\begin{matrix} (iv) & W_{0} (x) = 1 for x \end{matrix}$

Approximating the p-series

For $x \in R$ and $k \in N_{0},$ the upper (rising) Pochhammer [7] product x^(k) is defined as $x^{(0)} : = 1 and x^{(k)} : = \prod_{i = 0}^{k - 1} (x + i) = x (x + 1) \cdot \dots \cdot (x + k - 1) (k \in N) .$

For $p \in R,$ $q \in N$ and $x \in R^{+}$ we define the auxiliary sums $σ_{1} (x, p) : = 0 and σ_{q} (x, p) = \sum_{i = 1}^{q - 1} \frac{B_{2 i} \cdot p^{(2 i - 1)}}{x^{p + 2 i - 1} \cdot (2 i)!} (q \in N ∖ {1}) .$

The approximation of the nth harmonic number (1) is given in the following two theorems where the parameters m and q control the error R_q(m, n, p).

Theorem 1

Bounding harmonic numbers H_n

For positive integers m, n and q, where m < n, we have $\begin{matrix} H_{n} : = H (n, 1) = A_{q} (m, 1) + B_{q} (n, 1) + R_{q} (m, n, 1), \end{matrix}$ where⁴

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Convergence of p-series revisited with applications
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Y. Hansheng et al.
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There are more references available in the full text version of this article.

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Accurate double inequalities for generalized harmonic numbers

Abstract

Introduction

Section snippets

Euler–Maclaurin summation formula

Approximating the p-series

Bounding harmonic numbers Hn

Convergence of p-series revisited with applications

Int. J. Math. Math. Sci.

Handbook of Mathematical Functions

An approximate formula for a partial sum of the divergent p-series

Appl. Math. Lett.

Another proof for the p-series test

College Math. J.

Bounding harmonic numbers H_n