HW03 – gilbert – (55035)
1
This
printout
should
have
18
questions.
(ii) while if
Multiplechoice
questions
may
continue
on
X
k
before
thenext
answering.
column
or
page
–
find
all
choices
0
<
c
k
≤
b
k
,
c
k
converges,
then the Comparison Test is inconclusive be
001
10.0 points
cause
X
b
k
could converge,
but
it
could
di
If a
k
,
b
k
, and
c
k
satisfy the inequalities
verge  we can’t say precisely without further
restrictions on b
k
.
0
<
a
k
≤
c
k
≤
b
k
,
Consequently, what we can say is
for all k, what can we say about the series
(A) converges ,
(B) need not converge .
∞
∞
(A) :
k
X
=1
a
k
,
(B) :
k
X
=1
b
k
002
10.0 points
Determine whether the series
if we know that the series
∞
2
n
X
=1
∞
3
n(n +
1)(n + 5)
(C) :
k
X
=1
c
k
p
converges or diverges.
is convergent but know nothing else about a
k
1.
series is convergent
and b
k
?
1.
(A) diverges ,
(B) converges
2.
series is divergent correct
Explanation:
Note first that
2.
(A) need not converge ,
(B) converges
n
3
lim
=
1
>
0 .
3.
(A) converges ,
(B) need not converge
n
→ ∞
n(n
+ 1)(n
+ 5)
correct
Thus by the limit comparison test, the given
series
2
4.
(A) converges ,
(B) converges
n
X
=∞
p
5.
(A) converges ,
(B) diverges
converges if and only if the series
∞
2
6.
(A) diverges ,
(B) diverges
n
X
=1
n
Explanation:
converges.
But by the pseries test with p = 1
Let’s try applying the Comparison Test:
(or use the comparison test applied to the har
(i) if
monic series), this last series diverges.
Conse
X
k
quently, the given
0
<
a
k
≤
c
k
,
c
k
converges,
series is divergent
.
then
the
Comparison
Test
applies
and
says
that
X
a
k
converges;
003
10.0 points
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2
Determine whether the series
∞
X
k
∞
₃
₃
n
X
3
(C)
4
n
= 15
k
= 1
(k + 2)3
k
converge(s)?
converges or diverges.
1. series is divergent
2. series is convergent correct
Explanation:
We use the Limit Comparison Test with
k
1
a
k
=
,
b
k
=.
(k + 2)3
k
3
k
For
lim
a
k
=
lim
k
=
1
>
0 .
k
→ ∞
b
k
k
→∞
k
+ 2
Thus the series
∞
k
X
=1
(k
+
k
2)3
k
converges if and only if the series
∞
X
1
k
= 1
3
k
converges. But this last series is a
geometric series with
r
=
1
3
<
1 ,
hence
convergent.
Consequently,
the
given
series is
series is convergent
.
004
10.0 points
Which of the following series
∞
(A)
n
X
=1
3n
2
2
n
+ 6
(B)
∞
₃
5
2
₃
n
n
X
=1
1. A and B only
2. A,
B,
and C
3. B and C only
correct
4. C only
5. B only
Explanation:
(A) Because of the way the n
TH
term is
defined as a quotient of polynomials in
the series, use of the integral test is
suggested. Set
2x
f (x)
=
3x
2
+ 6
.
Then f is continuous, positive and
decreasing on [1, ∞); thus
∞
X
2n
n
= 1
3n
2
+ 6
converges if and only if the improper integral
∞
2
x
1
3x
2
+ 6
dx
converges, which requires us to
evaluate the integral
n
2
x
I
n
=
1
3x
2
+ 6
dx .
Now after substitution (set u = x
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 Spring '07
 Sadler
 Mathematical Series

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