Exam 2 Solutions
1. False. For example, the odd terms of the alternating harmonic series
∞
X
n
=1
(

1)
n

1
n
form the
series
∞
X
k
=1
1
2
k

1
which diverges by the Integral Test.
2. (i)
s
1
=
1
2
,
s
2
=
5
6
,
s
3
=
23
24
, and
s
4
=
119
120
. It seems that
s
n
=
(
n
+1)!

1
(
n
+1)!
= 1

1
(
n
+1)!
.
(ii) Base step holds: 1

1
(1+1)!
= 1

1
2
=
1
2
=
s
1
. Now assume
s
m
= 1

1
(
m
+1)!
for some
m
≥
1.
Then we have
s
m
+1
=
s
m
+
m
+ 1
(
m
+ 1 + 1)!
=
±
1

1
(
m
+ 1)!
²
+
m
+ 1
(
m
+ 2)!
= 1

1
(
m
+ 2)!
and the induction proof is complete.
(iii) Since
s
n
= 1

1
(
n
+1)!
for all
n
≥
1, it’s clear that
s
n
→
1 as
n
→ ∞
. Hence
∞
X
n
=1
n
(
n
+ 1)!
= 1.
3. (i) Diverges: the even terms of the series converge to 1 and therefore the sequence of terms
doesn’t converge to 0.
(ii) Converges: use the Integral Test.
(iii) Converges: use the Ratio Test.
(iv) Converges: use the Root Test.
4. First note that
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 Fall '09
 WYNNE
 Harmonic Series, Mathematical Series, monotone decreasing

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