Abstract
In this article, we deduce a new family of upper bounds of
n
! of the form
n
!
<
2
π
n
(
n
/
e
)
n
e
M
n
[
m
]
n
∈
ℕ
,
M
n
[
m
]
=
1
2
m
+
3
1
4
n
+
∑
k
=
1
m
2
m
-
2
k
+
2
2
k
+
1
2
-
2
k
ζ
(
2
k
,
n
+
1
/
2
)
m
=
1
,
2
,
3
,
.
.
.
.
We also proved that the approximation formula
2
π
n
(
n
/
e
)
n
e
M
n
[
m
]
for big factorials has a speed of convergence equal to
n
-2
m-
3
for
m
= 1,2,3,..., which give us a superiority over other known formulas by a suitable choice of
m
.
Mathematics Subject Classification (2000)
: 41A60; 41A25; 57Q55; 33B15; 26D07.