series homework cal 2
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mth_162__review_3_s_2016.docMTH 162 N. Agras Unit 3 Review Spring 2016 _______________________ _____ PRINT LAST, First Name section
NO INK!!! WORK THAT IS NOT NEAT WILL NOT BE GRADED!!!!
A. Basic definitions and concepts
1. Definition: A sequence
}
{
n
a
converges means ______________________2. Definition: A series converges if
}
{
lim
n
n
s
¥
®
_______________3. A geometric series converges provided _____________________ and if a
geometric series converges, its sum
=
¥
S
4. A p-series converges provided ___________________________
5. To use the Integral Test on
å
¥
=
1
n
n
a
withn
a
n
f
=
)
(
these conditions are necessary:6. We use the Ratio test when the generator consists of ___________________
7. The nth Term test is used to prove ___________
8. The Harmonic series _______________ while the alternating Harmonic series
____________
9-14: Assume series of positive terms.
9.
å
¥
=
1
n
n
a
has the same order of magnitude aså
¥
=
1
n
n
b
if ___________________10. We use the limit comparison test when the generator consists of ______________
11. If
å
¥
=
1
n
n
b
converges andn
n
b
a
£
what can we conclude aboutå
¥
=
1
n
n
a
?12. If
å
¥
=
1
n
n
b
converges andn
n
b
a
³
what can we conclude aboutå
¥
=
1
n
n
a
?13. If
å
¥
=
1
n
n
b
diverges andn
n
b
a
³
what can we conclude aboutå
¥
=
1
n
n
a
?14. If
å
¥
=
1
n
n
b
diverges andn
n
b
a
£
what can we conclude aboutå
¥
=
1
n
n
a
?15. If
å
¥
=
1
n
n
b
converges andn
n
n
b
a
)
1
(
-
=
what can we conclude aboutå
¥
=
1
n
n
a
?16. If
å
¥
=
1
n
n
b
diverges andn
n
n
b
a
)
1
(
-
=
what can we conclude aboutå
¥
=
1
n
n
a
?17. The conditions that must be met for the alternating series
å
¥
=
1
n
n
a
EMBED Equation.3å
¥
=
-
=
1
)
1
(
n
n
n
b
to converge are:B. Determine if the sequence
{
}
n
a
converges or diverges. If it converges, find its limit. If it diverges, state the reason.1.
)
arctan(
n
a
n
=
______2.
÷
ø
ö
ç
è
æ
=
n
a
n
1
cos
______3.
(
)
n
a
n
p
cos
=
______4.
12
7
3
5
2
2
2
+
+
+
+
=
n
n
n
n
a
n
______5.
1
6
3
2
)
1
(
+
+
-
=
n
n
a
n
n
______6.
1
6
3
2
)
1
(
2
+
+
-
=
n
n
a
n
n
______7.
n
n
a
)
7
.
0
(
12
=
______8.
n
n
a
)
7
.
1
(
12
=
______C. Identify each of the following as a p-series, a multiple of a p-series, or a geometric series. Identify p or r. State whether the series converges or diverges.
1.
n
n
å
¥
=
÷
ø
ö
ç
è
æ
0
5
3
4
______2. 0.272727… ______
3.
(
)
n
n
å
¥
=
0
01
.
1
2
______4.
å
¥
=
1
2
3
1
n
n
______5.
...
25
1
16
1
9
1
4
1
1
+
+
+
+
______6.
...
7
1
6
1
5
1
2
1
3
1
2
1
+
+
+
+
+
+
______7.
...
625
8
125
4
25
2
5
1
+
+
+
______D. 1. Prove that
å
¥
=
1
n
n
a
has the same order of magnitude aså
¥
=
1
n
n
b
then state whether the series both converge or diverge.a)
å
¥
=
1
n
n
a
=å
¥
=
+
+
+
+
1
2
7
3
11
2
5
3
8
n
n
n
n
n
å
¥
=
1
n
n
b
=å
¥
=
1
4
1
n
n
b)
å
¥
=
1
n
n
a
=å
¥
=
+
+
+
1
3
7
5
2
8
1
n
n
n
n
å
¥
=
1
n
n
b
=å
¥
=
1
3
4
1
n
n
2. Find a p-series that has the same order of magnitude as the given series. Then state whether both the given series and its associated p-series converge or diverge.
a)
å
¥
=
+
+
+
1
3
2
3
5
4
n
n
n
n
b)
å
¥
=
+
+
+
1
5
1
5
3
2
n
n
n
n
c)
å
¥
=
+
+
1
2
5
1
3
n
n
n
E. Determine if the series converges or diverges and give the reason. Find the sum where possible.
1.
...
27
1
8
1
9
1
4
1
3
1
2
1
+
+
+
+
+
+
____________________________________2.
å
¥
=
+
+
1
)
2
)(
1
(
2
n
n
n
____________________________________3.
å
¥
=
1
4
n
n
____________________________________4.
å
¥
=
-
1
4
)
1
(
n
n
n
____________________________________5.
å
¥
=
+
-
1
2
)
1
(
n
n
n
____________________________________6.
å
¥
=
+
+
+
1
3
2
3
5
4
n
n
n
n
____________________________________7.
å
¥
=
+
+
1
3
2
3
1
n
n
n
____________________________________8.
å
¥
=
+
-
0
1
3
5
2
n
n
n
n
____________________________________9.
å
¥
=
+
-
0
1
6
5
2
n
n
n
n
____________________________________F. For each of the following, PROVE that the series converges or diverges using an appropriate test. Show each step of the process neatly.
1.
å
¥
=
1
3
3
n
n
n
2.
å
¥
=
+
+
0
2
1
sin
1
n
n
n
3.
å
¥
=
+
+
2
1
cos
2
n
n
n
4.
å
¥
=
+
1
1
2
n
n
n
n
5.
å
¥
=
+
+
1
2
1
3
n
n
n
ne
6.
å
¥
=
1
)!
2
(
n
n
n
n
7.
å
¥
=
+
+
+
0
2
2
7
5
3
5
n
n
n
n
8.
å
¥
=
2
ln
1
n
n
n
9.
å
¥
=
+
+
-
0
2
1
2
1
)
1
(
n
n
n
n
10.
å
¥
=
0
2
arctan
n
n
n
11.
å
¥
=
2
4
ln
n
n
n
G. Compute the first four terms, to five decimal places, of the sequence of partial sums for each of the following. State whether each series converges or diverges.
1. The Harmonic Series.
2. p=3
3.
å
¥
=
0
cos
n
n
p
H. Test for absolute convergence. If the series does not converge absolutely, then test for conditional convergence.
1.
å
¥
=
+
-
1
4
)
1
(
n
n
n
2.
(
)
å
¥
=
+
-
0
2
2
1
n
n
n
3.
(
)
å
¥
=
+
-
1
1
3
2
1
n
n
n
n
n
I. Power Series. Find the interval of convergence for each of the following power series. Don’t forget to test the endpoints!
1.
å
¥
=
-
1
1
3
n
n
n
x
____________2.
å
¥
=
+
1
1
2
n
n
n
x
____________3.
å
¥
=
+
-
1
1
3
)
2
(
n
n
n
x
n
____________4.
å
¥
=
+
-
-
1
1
)
3
(
)
1
(
n
n
n
n
x
____________5.
å
¥
=
-
-
1
10
)
1
(
!
)
1
(
n
n
n
n
x
n
____________6.
å
¥
=
-
-
1
)
1
(
)
1
(
n
n
n
n
n
x
____________J. Use the power series expansions for
x
-
1
1
and forx
+
1
1
to obtain the power series for each of the following. Note: All power series expansions should be expressed in generator form.1.
)
1
ln(
x
-
2.
2
3
1
x
x
+
3.
x
+
2
1
4.
(
)
5
1
tan
x
-
K. Given the power series expansions found on your formula sheet, give the power series expansion in generator form for each.
1.
x
x
cos
2
__________________2.
2
sin
x
__________________3.
dx
e
x
x
3
3
ò
__________________4.
dx
x
x
ò
2
4
cos
__________________5.
dx
x
x
ò
cos
__________________6. Estimate
2
.
0
cos
using4
T
with a = 0 __________7. Estimate
2
.
0
e
using3
T
with a = 0 __________8. Estimate
dx
x
ò
4
.
0
0
2
sin
to four decimal places __________________9. Expand
x
e
in a Maclaurin series by definition. Determine the values of x for which the series converges.10. Expand
x
cos
in a Taylor series aboutp
=
a
by definition. Determine the values of x for which the series converges.11. Expand
x
sin
in a Taylor series about2
p
=
a
by definition. Determine the values of x for which the series converges.12. Find
3
T
with a = 1 for the functionx
x
x
f
ln
)
(
=
