moseley (cmm3869) – HW04 – Gilbert – (56380)
1
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001
10.0 points
Determine whether the series
∞
summationdisplay
n
=0
4 (3)
n
is convergent or divergent, and if convergent,
find its sum.
1.
convergent, sum =
−
5
2
2.
convergent, sum =
5
2
3.
convergent, sum = 1
4.
divergent
correct
5.
convergent, sum =
−
2
Explanation:
The given series is an infinite geometric
series
∞
summationdisplay
n
=0
a r
n
with
a
= 4 and
r
= 3. But the sum of such a
series is
(i) convergent with sum
a
1
−
r
when

r

<
1,
(ii) divergent when

r
 ≥
1.
Consequently, the given series is
divergent
.
002
10.0 points
Find the sum of the infinite series
∞
summationdisplay
k
=1
(cos
2
θ
)
k
,
(0
≤
θ <
2
π
)
,
whenever the series converges.
1.
sum = cot
2
θ
correct
2.
sum = csc
2
θ
3.
sum = tan
2
θ
4.
sum = sec
2
θ
5.
sum = sin
2
θ
cos
2
θ
Explanation:
For general
θ
the series
∞
summationdisplay
k
=1
(cos
2
θ
)
k
is an infinite geometric series with common
ratio cos
2
θ
. Since the series starts at
k
= 1,
its sum is thus given by
cos
2
θ
1
−
cos
2
θ
=
cos
2
θ
sin
2
θ
.
Consequently
sum = cot
2
θ
.
003
10.0 points
Determine whether the series
∞
summationdisplay
n
=1
n
3
5
n
3
+ 3
is convergent or divergent, and if convergent,
find its sum.
1.
convergent with sum = 8
2.
convergent with sum =
1
5
3.
convergent with sum =
1
8
4.
convergent with sum = 5
5.
divergent
correct
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moseley (cmm3869) – HW04 – Gilbert – (56380)
2
Explanation:
The infinite series
∞
summationdisplay
n
=1
a
n
is divergent when
lim
n
→∞
a
n
exists but
lim
n
→∞
a
n
negationslash
= 0
.
Note for the given series,
a
n
=
n
3
5
n
3
+ 3
=
1
5 +
3
n
3
,
so
lim
n
→∞
a
n
= lim
n
→∞
n
3
5
n
3
+ 3
=
1
5
negationslash
= 0
.
Thus the given series is
divergent
.
004
10.0 points
Determine if the series
∞
summationdisplay
n
=1
2 + 3
n
4
n
converges or diverges, and if it converges, find
its sum.
1.
converges with sum = 3
2.
converges with sum =
8
3
3.
series diverges
4.
converges with sum =
7
3
5.
converges with sum =
11
3
correct
6.
converges with sum =
10
3
Explanation:
An infinite geometric series
∑
∞
n
=1
a r
n
−
1
(i) converges when

r

<
1 and has
sum =
a
1
−
r
,
while it
(ii) diverges when

r
 ≥
1
.
Now
∞
summationdisplay
n
=1
2
4
n
=
∞
summationdisplay
n
=1
1
2
parenleftBig
1
4
parenrightBig
n
−
1
is a geometric series with
a
=
r
=
1
4
<
1.
Thus it converges with
sum =
2
3
,
while
∞
summationdisplay
n
=1
3
n
4
n
=
∞
summationdisplay
n
=1
3
4
parenleftBig
3
4
parenrightBig
n
−
1
is a geometric series with
a
=
r
=
3
4
<
1.
Thus it too converges, and it has
sum = 3
.
Consequently, being the sum of two conver
gent series, the given series
converges with sum =
2
3
+ 3 =
11
3
.
005
10.0 points
Determine whether the infinite series
∞
summationdisplay
n
=1
tan
−
1
parenleftBig
3
n
2
4
n
+ 1
parenrightBig
converges or diverges, and if it converges, find
its sum.
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 Spring '07
 Sadler
 Multivariable Calculus, Mathematical Series, lim, Moseley

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