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In-Depth Information
The evaluation of
b
(
n
)
is strictly related to the following Wallis' product for
π
/
2(a
consequence of
Euler's jewel
given in Sect. 5.4.3):
4
n
2
4
n
2
4
i
2
4
i
2
=
n
>
0
→
∞
∏
i
π
/
2
1
=
lim
n
(7.5)
−
−
1
=
1
,
n
2
=
2
1
·
2
3
·
4
3
·
4
5
·
6
5
·
6
7
... ·
2
n
2
n
2
n
2
n
1
·
1
...
(7.6)
−
+
if we denote by
(
2
n
)
!! the product of the even numbers equal to or less than 2
n
,we
get:
2
n
(
2
n
)
!!
=
·
n
!
(7.7)
moreover, if we denote by
(
2
n
+
1
)
!! and by
(
2
n
−
1
)
!! the product of the odd num-
bers equal to or less than 2
n
+
1and2
n
−
1 respectively, we get:
2
n
(
2
n
−
1
)
!!
=(
2
n
)
!
/
(
·
n
!
)
(7.8)
2
n
(
2
n
+
1
)
!!
=(
2
n
+
1
)
·
(
2
n
)
!
/
(
·
n
!
)
(7.9)
therefore, Eq. (7.6) becomes:
(
2
n
)
!!
·
(
2
n
)
!!
π
/
2
→
n
→
∞
!!
whence, according to Eqs. (7.7), (7.8), and (7.9), it follows that:
(
2
n
+
1
)
!!
·
(
2
n
−
1
)
2
n
n
!
2
n
n
!
·
π
/
2
→
n
→
∞
(
2
n
+
1
)
·
(
2
n
)
!
/
(
2
n
·
n
!
)
·
(
2
n
)
!
/
(
2
n
·
n
!
)
that is:
2
4
n
4
2
→
n
→
∞
=
(
n
!
)
)
.
2
((
2
n
)
!
)
(
2
n
+
1
2
,
b
2
Now, let us consider the terms
(
b
(
n
))
(
2
n
)
and evaluate the ratio
(
b
(
n
))
/
b
(
2
n
)
.
We h ave :
2
e
2
n
n
2
n
=
(
)
n
!
2
(
b
(
n
))
!
e
2
n
)=
(
2
n
)
(
b
2
n
(
2
n
)
2
n
2
2
e
2
n
n
2
n
2
n
2
2
2
n
n
2
n
(
2
2
n
2
(
b
(
n
))
)
=
(
n
!
)
(
2
n
)
!
e
2
n
=
(
n
!
)
(
n
!
)
!
=
n
2
n
b
(
2
n
(
2
n
)
2
n
)
(
2
n
)
!
1
/
2
by
1
/
2
, becomes:
(
+
)
(
)
which, by approximating
2
n
1
2
n
2
→
n
→
∞
√
π
(
b
(
n
))
1
/
2
1
/
2
)
=(
2
n
+
1
)
(
π
/
2
)
n
.
b
(
2
n