A Sharp Double Inequality for Sums of Powers

Abstract

It is established that the sequences and are strictly increasing and converge to and , respectively. It is shown that there holds the sharp double inequality .

1. Introduction

The proof of the equality

(1.1a)

published recently in the form [1]

(1.1b)

was based on the equations with the false hypothesis that big is independent of (see [1, pages 63-64] and [2, pages 54-55]). Deriving (1.1b) the author used the Euler-Maclaurin summation formula and a generating function for the Bernoulli numbers.

Subsequently, Spivey published the correction of his demonstration as the Letter to the Editor [2]. Additionally, Holland [3] published two different derivations of (1.1a) in the same issue as Spivey's correction appeared.

In this note, using only elementary techniques, we demonstrate that the sequence is strictly increasing and that (1.1a) holds; in addition, we establish a sharp estimate of the rate of convergence.

2. Monotone Convergence

The formula (1.1a) is illustrated in Figure 1, where the sequence is depicted. Its monotonicity is seen very clearly.

Figure 1

The graph of the sequence .

To prove that the sequence is strictly increasing, we change the order of summation

(2.1)

Now, consider the function which is, for , strictly increasing on the open interval and , for any [4, page 42]. Consequently, the sequence is strictly increasing. We use Tannery's theorem for series (see [5] or [6, item 49, page 136]) to determine its limit.

Lemma 2.1 (Tannery).

Let a double sequence of complex numbers satisfy the following conditions:

(1)The finite limit exists for every fixed .

(2)There exists a sequence of positive constants such that for every satisfying the estimate , and the series converges. (In [6, item 49, page 136], we have the stronger supposition that for all ).

Then we have

(2.2)

Proof.

Let all the conditions of the Lemma be satisfied and be given. Then we estimate for and for some . Moreover, for any , also for at some . Thus, for , we estimate

(2.3)

Now, using (2.1) and putting and into Tannery's Lemma, we obtain

(2.4)

3. The Rate of Convergence

Referring to Figure 1, the convergence of the sequence appears to be rather slow. The difference

(3.1)

determines the sequence . Its graph, shown in Figure 2, suggests it is monotonic increasing, which we will prove first.

Figure 2

The graph of the sequence .

Indeed, according to (3.1) and (2.1), we have

(3.2)

where

(3.3)

and, for , the sequence is strictly decreasing and converges to zero [4, (4)]. Thus, we have

(3.4)

with

(3.5)

To examine the monotonicity of the sequence , we study, using (3.3), (3.4), and (3.5), the difference , which is equal to

(3.6)

Hence:

(3.7)

Next, we examine also the question of convergence of the above sequence. First, referring to (3.3), (3.5), and [4, page 29, equation (16)], there exists the limit

(3.8)

Moreover, according to (3.3), (3.5), and [4, (15)], the estimates

(3.9)

hold true for . Additionally, , due to (3.3) and (3.5). Thus, the estimate

(3.10)

is being valid for and with

(3.11)

According to (3.8) and differentiating the appropriate power series resulting from the geometric series, we obtain

(3.12)

Now, referring to (3.4) and (3.8)–(3.12), and applying Tannery's Lemma–-equation (2.2), with , we obtain the result

(3.13)

Therefore, using (3.1) and (3.7), we find the following sharp inequality

(3.14)

true for every . In addition, we have also the estimate

(3.15)

valid for every such that .

We have , and for the function we calculate , and . This way we obtain simple and rather accurate estimates

(3.16)

Consequently, we get, for example, a simple double inequality

(3.17)

Open Question 3.

Are the sequences and strictly concave?