CUBO, A Mathematical Journal

Vol. 23, no. 03, pp. 357–368, December 2021

DOI: 10.4067/S0719-06462021000300357

Basic asymptotic estimates for powers of Wallis’
ratios

Vito Lampret1

1 University of Ljubljana, Ljubljana, 386

Slovenia, EU.

vito.lampret@guest.arnes.si

ABSTRACT

For any a ∈ R, for every n ∈ N, and for n-th Wallis’ ratio
wn :=

∏n
k=1

2k−1
2k

, the relative error r0(a, n) :=
(
v0(a, n) −

wan
)
/wan of the approximation w

a
n ≈ v0(a, n) := (πn)

−a/2

is estimated as
∣∣r0(a, n)∣∣ < 14n . The improvement wan ≈

v(a, n) := (πn)−a/2
(
1 − a

8n
+ a

2

128n2

)
is also studied.

RESUMEN

Para cualquier a ∈ R, para todo n ∈ N, y para el n-
ésimo cociente de Wallis wn :=

∏n
k=1

2k−1
2k

, el error re-

lativo r0(a, n) :=
(
v0(a, n) − w

a
n

)
/wan de la aproximación

wan ≈ v0(a, n) := (πn)
−a/2 se estima como

∣∣r0(a, n)∣∣ <
1
4n

. También se estima la mejora wan ≈ v(a, n) :=
(πn)−a/2

(
1 − a

8n
+ a

2

128n2

)
.

Keywords and Phrases: approximation, asymptotic, estimate, inequality, power, Wallis’ ratio.

2020 AMS Mathematics Subject Classification: 41A60, 65B10, 11Y60, 33E20, 33F05, 40A25.

Accepted: 09 July, 2021

Received: 16 January, 2021

©2021 V. Lampret. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.4067/S0719-06462021000300357
https://orcid.org/0000-0002-2921-3451


358 Vito Lampret CUBO
23, 3 (2021)

1 Introduction

The sequence of Wallis1 ratios

wn :=

n∏
k=1

2k − 1
2k

=
(2n − 1)!!
(2n)!!

= 4−n
(
2n

n

)
(1.1)

is often encountered in pure and applied mathematics, in physics, and in several other exact sciences

too. For example, the perimeter P(a, b) of an ellipse having semi-axes of length a and b ≤ a is
given as

P(a, b) = 4a

∫ π/2
0

√
1 − ε2 sin2(τ) dτ = 2aπ

(
1 −

∞∑
k=1

w2k
2k − 1

ε2k

)
(1.2)

[20], where ε =
√

1 − b2
a2
, the eccentricity of an ellipse.

Similarly, the period T of a simple pendulum, located in the gravitational field with the acceleration

g and having the length L and the amplitude of the oscillation α ∈ (0, π), is determined by the
formula

T = 4

√
L

g

∫ π/2
0

dτ√
1 − ε2 sin2(τ)

= 2π

√
L

g

(
1 +

∞∑
k=1

w2kε
2k

)
(1.3)

[21, p. 26], where ε = sin(α/2). Not only in mechanics, but also in other parts of physics, the

Wallis ratio has several interesting roles, see for example [9] and [12].

In mathematics, the sequence of the Landau constants Gn, important in the theory of analytic

functions, see [1], is also defined by the Wallis ratios as

Gn :=

n∑
k=1

w2k (n ∈ N). (1.4)

The Wallis ratio attracts mathematicians also because of its direct connections with Catalan num-

bers cn :=
1

n+1

(
2n
n

)
, also important objects for pure and applied mathematics [15, 29]. In fact, the

Wallis ratio, i.e. the sequence n 7→ wn, was investigated by many researches, see for example the
papers [2, 3, 4, 5, 6, 7, 8, 11, 14, 16, 22, 23, 26, 27, 28, 29, 31, 33].

In 2007 was presented [33] aesthetically pleasing double inequality

1
√
eπn

(
1 +

1

2n

)n− 1
12n

< wn ≤
1

√
eπn

(
1 +

1

2n

)n− 1
12n+16

, (1.5)

true for all n ∈ N.

In 2013 was demonstrated [10] the estimate√
e

π

(
1 −

1

2n

)n √
n − 1
n

< wn ≤
4

3

(
1 −

1

2n

)n √
n − 1
n

, (1.6)

1John Wallis, 1616 – 1703



CUBO
23, 3 (2021)

Basic asymptotic estimates for powers of Wallis’ ratios 359

true for n ≥ 2.

In 2015 was derived [11] the inequalities(
2

3

)3/2 (
1 −

1

2n

)n+1/2 (
n −

3

2

)−1/2
≤ wn <

√
e

π

(
1 −

1

2n

)n+1/2 (
n −

3

2

)−1/2
, (1.7)

valid for n ≥ 2. At the same time, in [28, Theorems 4.2 and 5.2] were presented the estimates

wn >

√
e

πn

(
1 −

1

2n

)n
exp

(
1

24n2
+

1

48n3
+

1

160n4
+

1

960n5

)
(1.8)

wn >

√
e

πn

(
1 −

1

2(n + 1/3)

)n+1/3
(1.9)

and

wn <

√
e

πn

(
1 −

1

2(n + 1/3)

)n+1/3
exp

(
1

144n3

)
, (1.10)

all true for n ≥ 1.

In the mentioned formulas for the perimeter of an ellipse and the period of a simple pendulum, as

well as for the Landau sequence, see (1.2)–(1.4), we met the second powers of the Wallis ratios.

This fact initiated our desire to approximate any power of the Wallis ratio. But, all the inequalities

(1.5)–(1.10) are less suitable for estimating the power wan for a ∈ R. Fortunately, the approximation
formula for the Wallis ratio, presented in [19], is more convenient for this task. In this contribution

we shall show the first two steps how to approximate simply and accurately the powers of the

Wallis ratios having real exponents.

2 Basic discussion

The sequence of Wallis’ ratios was estimated recently [19] as

wn =
1

√
π n

exp
(
− s̃r(n) + δr(n)

)
(n ∈ N), (2.1)

where

s̃r(n) =

r∑
i=1

(1 − 4−i)B2i
i(2i − 1)n2i−1

(n, r ∈ N) (2.2)

and, for any n, r ∈ N, the error δr(n) is estimated as

−
∣∣B2r+2∣∣

(r + 1)(2r + 1)n2r+1
< (−1)rδr(n) <

∣∣B2r+2∣∣
2(r + 1)(2r + 1)(2n)2r+1

. (2.3)

Here B2 =
1
6
, B4 = − 130 , B6 =

1
42

, . . . are the Bernoulli numbers, defined by the identity

x
ex−1 ≡

∑∞
j=0 Bj

xj

j!
( |x| < 2π ).

We obtain the basic approximation by using r = 1,

wan = (πn)
−a/2 exp

(
− as̃1(n) + aδ1(n)

)
(a ∈ R, n ∈ N), (2.4)



360 Vito Lampret CUBO
23, 3 (2021)

with, for n ∈ N,

s̃1(n) :=
1

8n
> 0 (2.5)

and

−
1

180 n3
< −

1

2880 n3
< δ1(n) <

1

180 n3
. (2.6)

Thus, due to (2.5),

as̃1(n) =
a

8n
(a ∈ R, n ∈ N) . (2.7)

Moreover, thanks to (2.5)–(2.6), we estimate, for n ∈ N,

−s̃1(n) ±
∣∣δ1(n)∣∣ ≥ −s̃1(n) − ∣∣δ1(n)∣∣ > − 1

8n
−

1

180n3
> −

1

7n
(2.8)

and

−s̃1(n) ±
∣∣δ1(n)∣∣ ≤ −s̃1(n) + ∣∣δ1(n)∣∣ < − 1

8n
+

1

180n3
< −

1

9n
. (2.9)

Therefore, − a
7n

< a
(
− s̃1(n) ± δ1(n)

)
< − a

9n
, for a > 0 and − a

7n
> a

(
− s̃1(n) ± δ1(n)

)
> − a

9n
,

for a < 0. Thus,

min
{
− a

7n
, − a

9n

}
< a

(
− s̃1(n) ±

∣∣δ1(n)∣∣) < max{− a7n, − a9n} (a ̸= 0, n ∈ N). (2.10)
Hence, considering (2.4), together with the equality min(−S) = − max(S), for every S ⊆ R, we
derive the following theorem.

Theorem 2.1. For a ∈ R ∖ {0} and n ∈ N, the following double inequality holds:

(πn)−a/2 exp
(
− max

{
a
7n

, a
9n

})
< wan < (πn)

−a/2 exp
(
− min

{
a
7n

, a
9n

})
. (2.11)

Figure 1 shows2 the graphs of the function a 7→ wa2 and its approximation (dashed line) a 7→
(π · 2)−a/2.

-2 -1 1 2

1

2

3

4

5

6

7

n=2

-0.10 -0.05 0.05 0.10

0.95

1.00

1.05

1.10

n=2

Figure 1: The graphs of the function a 7→ wa2 and its approximation (dashed line) a 7→ (π · 2)−a/2.

2All graphics and calculations in this paper are made using the Mathematica [32] computer system.



CUBO
23, 3 (2021)

Basic asymptotic estimates for powers of Wallis’ ratios 361

Example 2.2. For any n ∈ N we have

(πn)−50 exp
(
− 100

7n

)
< w100n < (πn)

−50 exp
(
− 100

9n

)
,

(πn)50 exp
(
100
9n

)
< w−100n < (πn)

50 exp
(
100
7n

)
.

From Theorem 2.1 there follows the next corollary.

Corollary 2.3. For every a ∈ R ∖ {0} and for any positive integer n ≥ a we have

wan >
6

7
(πn)−a/2 . (2.12)

Proof. For k ≥ a > 0, using (2.11), we obtain3 wak > (πk)
−a/2 exp

(
− a

7k

)
> (πk)−a/2(1 −

a
7k
) ≥

(
πk)−a/2(1 − 1

7

)
= 6

7
(πk)−a/2. Furthermore, for a < 0, due to (2.11), we estimate wak >

(πk)−a/2 exp
(
− a

9k

)
= (πk)−a/2 exp

( |a|
9k

)
> (πk)−a/2 · 1.

Lemma 2.4. Let real numbers α, β, v and w satisfy the inequalities αβ ≥ 0, β ≤ 1
2
, v > 0 and

eαv < w < eβv. Then we have |v − w| < 3
2
v · max{|α|, |β|}.

Proof. Supposing that all conditions of Lemma 2.4 are satisfied, we have only two possibilities

α < β ≤ 0 or 0 ≤ α < β, together with the estimate

(eα − 1)v < w − v < (eβ − 1)v.

Therefore, in case α ≤ 0, we have (1 − eα)v > v − w > (1 − eβ)v ≥ 0. Thus, see Footnote 3,
|v−w| < −α = |α|. Additionally, using the first order Taylor’s formula and the estimate 0 ≤ β ≤ 1

2
,

in case α ≥ 0, we obtain, 0 ≤ (eα − 1)v < −(v − w) < (eβ − 1)v < β + 1
2
eββ2 ≤ β + 1

2
e1/2 1

2
β < 3

2
β.

Hence, in both cases we have |v − w| < v · max{|α|, 3
2
|β|}.

Corollary 2.5 (relative error). For every a ∈ R ∖ {0} and for any positive integer n ≥ a the
relative error r0(a, n) :=

(
wan − v0(a, n)

)
/wan of the approximation w

a
n ≈ v0(a, n) := (πn)−a/2 is

roughly estimated as ∣∣r0(a, n)∣∣ < 1
4n

.

Proof. Thanks to Theorem 2.1 and Lemma 2.4, using the notations α = − max
{

a
7n

, a
9n

}
, β =

− min
{

a
7n

, a
9n

}
, v = v0(a, n) = (πn)

−a/2 and w = wan, we obtain∣∣v0(a, n) − wan∣∣ < 32(πn)−a/2 · max
{∣∣max{ a

7n
, a
9n

}∣∣ , ∣∣min{ a
7n

, a
9n

}∣∣}.
Thus, according to the identity max

{
|max {x, y}| , |min {x, y}|

}
= max

{
|x|, |y|

}
, we get

∣∣v0(a, n) − wan∣∣ < 32(πn)−a/2 · |a|7n .
3considering the well-known estimate ex > 1 + x, true for x ∈ R ∖ {0}.



362 Vito Lampret CUBO
23, 3 (2021)

Hence, using Corollary 2.3,∣∣v0(a, n) − wan∣∣
wan

<
3

2
(πn)−a/2

|a|
7n

·
7

6
(πn)a/2 =

|a|
4n

.

Figure 2 shows, on the left – the graph of the actual relative error function a 7→ r0(a, n) and on
the right – the graphs of the functions a 7→ r0(a, n) and a 7→

|a|
4×1000 (dashed line).

-10 000 -5000 -2000 2000 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

a

b

b=r0Ha,1000L

-8 -6 -4 -2 2 4 6 8

0.0005

0.0010

0.0015

0.0020

a

b

b=r0Ha,1000L

b=
a

4´1000

Figure 2: Left – the graph of the actual relative error function a 7→ r0(a, 1000); Right – the graphs
of the actual relative error a 7→ r0(a, 1000) and its approximation (dashed line) a 7→

|a|
4×1000 .

3 Improvement

The relations (2.4)–(2.6) can be exploited more accurately to derive the next theorem.

Theorem 3.1. For any a ∈ R and every integer n ≥ |a|
8
, we have

wan = v(a, n) + ε(a, n), (3.1)

where

v(a, n) := (πn)−a/2
(
1 −

a

8n
+

a2

128n2

)
, (3.2)

and the error ε(a, n) is estimated as

∣∣ε(a, n)∣∣ ≤ ε∗(a, n) := (πn)−a/2 [ a2
100

+
1

18
exp

(
− min

{ a
7n

,
a

9n

})] |a|
10n3

(3.3)

≤ (πn)−a/2
(

a2

100
+

1

18
exp

(|a|
7n

)) |a|
10n3

≤ ε∗∗(a, n) := (πn)−a/2
(
a2 +

35

2

)
|a|

(10n)3
. (3.4)

Proof. Using the second order Taylor’s formula, we have

exp
(
− as̃1(n)

)
= exp

(
−

a

8n

)
= 1 −

a

8n
+

1

2

(
−

a

8n

)2
+ R2(a, n) (3.5)



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23, 3 (2021)

Basic asymptotic estimates for powers of Wallis’ ratios 363

with

R2(a, n) =
1

6
exp

(
−ϑ ·

a

8n

)(
−

a

8n

)3
, for some ϑ = ϑ(a, n) ∈ (0, 1).

Therefore, for a ∈ R and n ≥ |a|
8
,

∣∣R2(a, n)∣∣ ≤ 1
6
exp

(
|a|
8n

)
·
(
|a|
8n

)3
≤

e

6
·

|a|3

512n3
≤

|a|3

1000n3
. (3.6)

Similarly,

exp
(
aδ1(n)

)
= 1 + exp

(
ϑ · aδ1(n)

)
· aδ1(n), (3.7)

for some ϑ = ϑ(a, n) ∈ (0, 1).

Thanks to (3.7), (2.10) and (2.6), we estimate, using some θ = θ(a, n) ∈ (0, 1),∣∣∣exp( − as̃1(n) + aδ1(n)) − exp( − as̃1(n))︸ ︷︷ ︸
=∆(a,n)

∣∣∣ = exp( − as̃1(n)) · ∣∣exp(θ · aδ1(n)) · aδ1(n)∣∣
≤ exp

(
− as̃1(n)

)
· exp

(
|aδ1(n)|

)
·
∣∣aδ1(n)∣∣

= exp
(
a
(
− s̃1(n) ± |δ1(n)|

))
|a|
∣∣δ1(n)∣∣

(2.10)

≤
(2.6)

exp
(
max

{
−

a

7n
, −

a

9n

})
·

|a|
180n3

. (3.8)

Consequently, according to (2.4) and (3.5), we obtain

wan
(2.4)
= (πk)−a/2

(
exp

(
−as̃1(n)+aδ1(n)

)) (3.5)
= (πn)−a/2

(
1 −

a

8n
+

a2

128n2
+ R2(a, n)︸ ︷︷ ︸

=exp(−as̃1(n))

+∆(a, n)
)
,

where, considering (3.6) and (3.8), for a ∈ R and n ≥ |a|
8
, we estimate the error ε(a, n) :=

(πn)−a/2
(
R2(a, n) + ∆(a, n)

)
as

∣∣ε(a, n)∣∣ ≤ (πn)−a/2 [ |a|3
1000n3

+ exp
(
− min

{ a
7n

,
a

9n

}) |a|
180n3

]
= (πn)−a/2

[
a2

100
+

1

18
exp

(
− min

{ a
7n

,
a

9n

})] |a|
10n3

≤ (πn)−a/2
[
a2

100
+

1

18
exp

(|a|
8n

·
8

7

)] |a|
10n3

≤ (πn)−a/2
(

a2

100
+

1

18
exp

(
1 ·

8

7

)) |a|
10n3

≤ (πn)−a/2
(

a2

100
+

7

40

)
|a|
10n3

.

Remark 3.2. The sequence n 7→ Wn := 12n+1
(∏n

k=1
2k

2k−1

)2
, called the Wallis sequence, is closely

connected to the sequence of the Wallis ratios wn by the identity Wn = w
−2
n /(2n + 1). So, Wn can

be estimated easily using Theorem 3.1, e.g. its consequence (3.14).



364 Vito Lampret CUBO
23, 3 (2021)

Remark 3.3. According to Theorem 3.1, the constant π can be easily approximated using certain

rational functions R∓(n). For example, from (3.14) we get, for any n ∈ N,

1

n

(
w−2n −

1

5n2

)(
1 +

1

4n
+

1

32n2

)−1
< π <

1

n

(
w−2n +

1

5n2

)(
1 +

1

4n
+

1

32n2

)−1
.

Directly from Theorem 3.1 and Corollary 2.3, from (3.4) and (2.12), we read the next corollary.

Corollary 3.4 (relative error). For every a ∈ R and for any positive integer n ≥ |a| the relative
error of the approximation wan ≈ v(a, n),

r(a, n) :=
wan − v(a, n)

wan
, (3.9)

is a priori estimated as ∣∣r(a, n)∣∣ ≤ r∗(a, n) := (a2 + 13
2

)
7|a|

6(10n)3
. (3.10)

For any a ∈ R and all integers n ≥ |a| the rough estimate r∗(a, n) < 8.2 h holds true.

Figure 3 shows the graphs of the actual relative error functions a 7→ r(a, n), for n ∈ {10, 100}.

-10 -5 5 10

0.0001

0.0002

0.0003

a

b

b = rHa,10L

-100 -50 50 100

0.0001

0.0002

0.0003

a

b

b = rHa,100L

Figure 3: The graphs of the actual relative error functions a 7→ r(a, n) for n ∈ {10, 100} .

Figures 4–5 compare the actual relative error functions a 7→ r(a, n) and their approximations
a 7→ r∗(a, n), for n ∈ {1, 3, 10, 100}.

-1.0 -0.5 0.5 1.0

0.002

0.004

0.006

0.008

n=1

a

b

b = rHa,1L

b = r
*

Ha,1L

-3 -2 -1 1 2 3

0.0005

0.0010

0.0015

0.0020

n=3

a

b

b = rHa,3L

b = r
*

Ha,3L

Figure 4: The graphs of the actual relative error functions a 7→ r(a, n) and their approximations
a 7→ r∗(a, n), for n ∈ {1, 3} .



CUBO
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Basic asymptotic estimates for powers of Wallis’ ratios 365

-10 -5 5 10

0.0002

0.0004

0.0006

0.0008

n=10

a

b = rHa,10L

b = r
*

Ha,10L

-100 -50 50 100

0.0002

0.0004

0.0006

0.0008

0.0010

n=100

a

b

b = rHa,100L

b = r
*

Ha,100L

Figure 5: The graphs of the actual relative error functions a 7→ r(a, n) and their approximations
a 7→ r∗(a, n), for n ∈ {10, 100} .

Using a ∈ {1, −1, 2, −2, 1
2
, π, −2π} in Theorem 3.1, considering (3.1) and (3.4), we obtain several

inequalities for Wallis’ ratios, presented in the next corollary.

Corollary 3.5. For every4 positive integer n we have

1
√
π n

(
1 − 1

8n
+ 1

128n2

)
− 1

95 n7/2
< wn <

1
√
π n

(
1 − 1

8n
+ 1

128n2

)
+ 1

95 n7/2
, (3.11)

√
π n
(
1 + 1

8n
+ 1

128n2

)
− 1

30 n5/2
<

1

wn
<

√
π n
(
1 + 1

8n
+ 1

128n2

)
+ 1

30 n5/2
, (3.12)

1

π n

(
1 − 1

4n
+ 1

32n2

)
− 1

73 n4
< w2n <

1

π n

(
1 − 1

4n
+ 1

32n2

)
+ 1

73 n4
, (3.13)

(π n)
(
1 + 1

4n
+ 1

32n2

)
− 1

7 n2
<

1

w2n
< (π n)

(
1 + 1

4n
+ 1

32n2

)
+ 1

7 n2
, (3.14)

1
4
√
π n

(
1 − 1

16n
+ 1

512n2

)
− 1

150n13/4
<

√
wn <

1
4
√
π n

(
1 − 1

16n
+ 1

512n2

)
+ 1

150n13/4
, (3.15)

1

(π n)π/2

(
1 − π

8n
+ π

2

128n2

)
− 1

70n3+π/2
< wπn <

1

(π n)π/2

(
1 − π

8n
+ π

2

128n2

)
+ 1

70n3+π/2
, (3.16)

(π n)π
(
1 + π

4n
+ π

2

32n2

)
− 14 nπ−3 < w−2πn < (π n)

π
(
1 + π

4n
+ π

2

32n2

)
+ 14 nπ−3 . (3.17)

Remark 3.6. In case a > 0, the inequalities in Corollary 3.5 can be slightly improved using (3.3)

instead of (3.4). For example, due to (3.3), we have, for a ∈ {1, 2},

|ε(1, n)| ≤ ε∗(1, n) = (π n)−1/2
(

1
100

+ 1
18

· 1
)

1
10n3

< 1
270n7/2

|ε(2, n)| ≤ ε∗(2, n) = (π n)−1
(

1
25

+ 1
18

· 1
)

2
10n3

< 1
164n4

.

4For 1 ≤ n < |a| the inequalities are approved directly.



366 Vito Lampret CUBO
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