Deriving a Sum Derive Euler’s famous result that was mentioned in Section 9.3,
∑
n
=
1
∞
1
n
2
=
π
2
6
by completing each step.
(a) Prove that
∫
d
v
2
−
u
2
+
v
2
=
1
2
−
u
2
arctan
v
2
−
u
2
+
C
(b) Prove that
I
1
=
∫
0
2
/
2
∫
−
u
u
2
2
−
u
2
+
v
2
d
v
d
u
=
π
2
18
by using the substitution
u
=
2
sin
θ
.
(c) Prove that
I
2
=
∫
2
/
2
2
∫
u
−
2
−
u
+
2
2
2
−
u
2
+
v
2
d
v
d
u
=
4
∫
π
/
6
π
/
2
arctan
1
−
sin
θ
cos
θ
d
θ
by using the substitution
u
=
2
sin
θ
.
(d) Prove the trigonometric identity
1
−
sin
θ
cos
θ
=
tan
[
(
π
/
2
)
−
θ
2
]
.
(e) Prove that
I
2
=
∫
2
/
2
2
∫
u
−
2
−
u
+
2
2
2
−
u
2
+
v
2
d
v
d
u
=
π
2
9
.
(f) Use the formula for the sum of an infinite geometric series to verify that
∑
n
=
1
∞
1
n
2
=
∫
0
1
∫
0
1
1
1
−
x
y
d
x
d
y
.
(g) Use the change of variables
u
=
x
+
y
2
and
v
=
y
−
x
2
to prove that
∑
n
=
1
∞
1
n
2
=
∫
0
1
∫
0
1
1
1
−
x
y
d
x
d
y
=
I
1
+
I
2
=
π
2
6
.