s1
=
Series
B

Log
B
+
y

+
y
F
,
8
y,
¥
,8
<F
1
y

1
3y
3
+
1
5y
5

1
7y
7
+
O
B
1
y
F
17 2
c) use this result to sum the series 1
1
3
+
1
5
+
.. and 1

1
3 3
+
1
5 3 3

1
7 3 3 3
+... . Confirm your results using
Sum.
Evaluating the series s1 at y > 1 generates the sum 1
1
3
+
1
5
+
.. , s1=ArcCot[y], so the sum is ArcCot[1]
ArcCot
@
1
D
Π
4
The formula for the nth term is
Table
@H

1
L
^
HH
n
LL
HH
2n
+
1
LL
,
8
n,0,5
<D
:
1,

1
3
,
1
5
,

1
7
,
1
9
,

1
11
>
Using Sum, we get
Sum
@H

1
L
^
HH
n
LL
HH
2n
+
1
LL
,
8
n,0,
¥
<D
Π
4
the same as using the ArcCot series.
Next, we note that the series 1

1
3 3
+
1
5 3 3

1
7 3 3 3
is
3
H
s1/.y>
3 ) , so the sum should be
Sqrt
@
3
D
ArcCot
@
Sqrt
@
3
DD
Π
2
3
Verifying with sum, the nth term is
Table
@H

1
L
^
HH
n
LL
H
3^
H
n
L H
2n
+
1
LL
,
8
n,0,6
<D
:
1,

1
9
,
1
45
,

1
189
,
1
729
,

1
2673
,
1
9477
>
so the sum is
Sum
@H

1
L
^
HH
n
LL
H
3^
H
n
L H
2n
+
1
LL
,
8
n,0,
¥
<D
2
212HomeWork2sol.nb