Start with
1
1 +
x
= 1 −
x
+
x
2
−
x
3
+
x
4
−
x
5
+ …
Replacing
x
with
x
2
yields
1
1 +
x
2
= 1 −
x
2
+
x
4
−
x
6
+
x
8
−
x
10
+ …
Integrating the both sides of the above gives
arctan
x
=
x
−
x
3
3
+
x
5
5
−
x
7
7
+
x
9
9
− …
Sub
x
= 1 in the above gives
arctan 1 = 1 −
1
3
+
1
5
−
1
7
+
1
9
− …
or
π
4
= 1 −
1
3
+
1
5
−
1
7
+
1
9
− …
File translated from T
E
X by
T
T
H
, version 4.03.
On 13 Jun 2022, 15:39.