sr
=
Series
@
denom,
8
r,0,7
<D
Normal
1
+
r
+
r
2
+
r
3
+
r
4
+
r
5
+
r
6
+
r
7
for clarity, convert this into a list :
srl
=
List
sr
9
1,r,r
2
,r
3
,r
4
,r
5
,r
6
,r
7
=
so the terms of the sum are :
int21
= ª

z
z
3
H
srl
L
.r
fi ª

z
9
ª

z
z
3
,
ª

2z
z
3
,
ª

3z
z
3
,
ª

4z
z
3
,
ª

5z
z
3
,
ª

6z
z
3
,
ª

7z
z
3
,
ª

8z
z
3
=
Now integrate them term by term :
int22
=
Map
@
Integrate
@
,
8
z,0,
¥
<D
&,int21
D
:
6,
3
8
,
2
27
,
3
128
,
6
625
,
1
216
,
6
2401
,
3
2048
>
pull out a factor of 6
int22 6
Expand
:
1,
1
16
,
1
81
,
1
256
,
1
625
,
1
1296
,
1
2401
,
1
4096
>
and the series is recognizable as
1
n
4
. So the value of the integral is 6
Ζ
[4] . Lets check this:
Integrate
B
z
3
ª
z

1
,
8
z,0,
¥
<F
Π
4
15
junk4sol.nb
3