Information Technology Reference
In-Depth Information
1
1
1
1
x
a
1
x
a
2
x
a
3
x
a
n
P
(
x
)=
−
−
−
...
−
Euler assumed that this property extends to “infinite polynomials” and applied it to
sin
x
/
x
. Namely, sin
x
/
x
is equal to zero in correspondence to the following values
of
x
:
π
,−
π
,
2
π
,−
2
π
,
3
π
,−
3
π
...
whence:
1
1
1
1
x
2
3!
+
x
4
5!
−
x
6
7!
+
x
8
9!
...
=
x
π
x
−
π
x
2
x
−
−
−
−
−
−
...
1
π
2
π
that is:
1
1
1
x
2
3!
+
x
4
5!
−
x
6
7!
+
x
8
9!
...
=
x
2
π
x
2
4
x
2
9
1
−
−
−
−
...
2
2
2
π
π
or equivalently:
1
π
x
2
x
2
3!
+
x
4
5!
−
x
6
7!
+
x
8
9!
...
=
1
1
1
−
1
−
2
+
2
+
2
+
...
...
4
π
9
π
therefore, by equating the coefficients of
x
2
of the two members (the other powers
do not matter, in this context), we get:
1
1
3!
=
−
1
π
1
4
+
1
9
+
1
16
+
...
+
2
thereby concluding that:
2
6
.
1
4
+
1
9
+
16
+
...
=
π
1
1
+
The sin
x
x
representation, in terms of its infinite roots, provides another important
result due to Wallis, which we use to prove Stirling's approximation (see Sect. 7.5).
In fact:
/
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
sin
(
π
/
2
)
−
π
/
−
π
/
−
π
/
−
π
/
2
=
...
2
2
2
2
π
/
π
4
π
9
π
16
π
that is:
π
=
(
2
4
−
1
)
(
16
−
1
)
(
36
−
1
)
...
4
16
36
which is exactly Wallis' identity, usually given by the following equality:
2
π
=
1
·
3
·
3
·
5
·
5
·
7
·
7
·
9
...
...
.
2
·
2
·
4
·
4
·
6
·
6
·
8
·
8
