MthSc 108 – 9.6 Solutions
(Alternating Series, Absolute and Conditional Convergence)
1.
Use the alternating series test to show that the series
(

1)
n
+
1
ln
n
n
n
=
1
∞
∑
converges.
The alternating series test applies because we have an alternating series

1
( )
n

1
b
n
n
=
1
∞
∑
=
b
1

b
2
+
b
3

b
4
+
b
5

b
6
+
L
where
b
n
>
0
.
Identify
b
n
:

1
( )
n

1
ln
n
n
=
ln
n
n
Let
f
(
x
)
=
ln
x
x
.
a. Determine if
lim
n
→∞
b
n
=
0
: Using L’Hospitals
lim
x
→∞
ln
x
x
=
lim
x
→∞
1
x
=
0.
b. Find the integer
N
, if there is one, for which
b
n
+
1
≤
b
n
:
If
f
(
x
)
=
ln
x
x
then
f
'(
x
)
=
1

ln
x
x
2
<
0 when
x
>
e
, which means that
f(x)
is decreasing
for
x
>
e
, which implies that
b
n
+
1
≤
b
n
for
N
≥
3
.
Based on these two conditions, the series converges by the Alternating Series Test.
2. Answer the following questions using the convergent alternating series
(

1)
n
n
4
n
n
=
1
∞
∑
.
a.
Find the 4
th
partial sum of the series as an exact value.
s
4
=
(

1)
1
⋅
1
4
1
+
(

1)
2
⋅
2
4
2
+
(

1)
3
⋅
3
4
3
+
(

1)
4
⋅
4
4
4
= 
5
32
b.
How accurate is the approximation from part (a) for the sum of the series?
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 Spring '08
 Any
 Calculus, Mathematical Series, lim, series A

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