1 Introduction

The famous Wallis sequence Wn， defined by

∶={1，2，3，…}

(1)

has the limiting value

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(2)

established by Wallis in 1655; see [1， p. 68]. Several elementary proofs of this well-known result can be found in [2-4]. An interesting geometric construction that produces the above limiting value can be found in Myerson [5]. Many formulas exist for the representation of π， and a collection of these formulas is listed [6-7]. For more history of π see [8-11]. Some inequalities and asymptotic formulas associated with the Wallis sequence Wn can be found in [12-18]. For example，Chen and Paris in [13，Remark 2.1.] establish the following asymptotic expansions for the Wallis sequence Wn

The gamma function

Γ(x)=tx-1e-tdt (x>0)

is one of the most important functions in mathematical analysis and has applications in many diverse areas. The logarithmic derivative of Γ(x), denoted by ψ(x)=Γ′(x)/Γ(x), is called the psi (or digamma) function.

In this paper， we present continued fraction representation for Wn by making use of the fact that

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Based on the obtained result， we establish a double inequality for the Wallis sequence.

2 Lemmas

The following Lemmas are required in our present investigation.

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Lemma 2.1([19， 20]) If the sequence (λn)n∈N converges to zero and if there exists the following limit

，

with s>1， then

We write the difference tn-tn+1 as the following power series in n-1：

(3)

where Bn(n∈0∶=∪{0}) are the Bernoulli numbers defined by the following generating function

In particular， it follows from (3) that， for x>0，

(4)

where

where

3 Main results

Theorem 3.1 The Wallis sequence Wn has the following continued fraction

representation：

(5)

From (6)， we have

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(6)

Proof (Step 1) we define the sequence (tn)n∈N by

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Lemma 2.2 ( [21， Corollary 1]) Let m，n∈， Then for x>0，

according to Lemma 2.1， the four parameter a，b，c and d， which produce the fastest convergence of the sequence (tn)n∈ are given by

that is

The speed of convergence of the sequence (tn)n∈ is given by the order estimate n-5 as n→∞.

and

Using the same method as above， we obtain that the sequence (ηn)n∈ converges fastest only if

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We thus find that

The speed of convergence of the sequence (ηn)n∈ is given by the order estimate n-9 as n→∞. The proof is complete.

The formula (5) motivated us to establish Theorem 3.2.

Theorem 3.2 For every n∈，

(7)

and

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where

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Proof The inequality (7) is equivalent to

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(8)

In order to prove (8)， it suffices to show that

f(n)>0 and g(n)<0 for n∈，

and

(Step 2) We define the sequence sequence (tn)n∈ by

Using Stirling’s formula， we find that

Differentiating f(x) and using the second inequality in (4)， we find， for x≥3

where P10(x) is a polynominal of degree 10 with non-negative integer coefficients.

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We then obtain that

f′(x)<0， x≥3.

Direct computation yields

f(1)=4.2722×10-5， f(2)=1.7560×10-6， f(3)=1.9004×10-7.

Hence， the sequence {f(n)} is strictly decreasing for n≥1， and we have

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which means that the first inequality in (8) is valid for n∈.

Differentiating g(x) and using the first inequality in (4)， we find， for x≥5，

where P9(x) is a polynomial of degree 9 with non-negative integer coefficients.

We then obtain that

g′(x)>0， x≥5.

Direct computation yields

Hence， the sequence {g(n)} is strictly increasing for n≥1， and we have

which means that the second inequality in (8) is valid for n∈. The proof is complete.

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