1220-90 Exam 2
Fall 2013
Name
Y
Instructions.
Show all work and include appropriate explanations when necessary. A correct answer
without accompanying work may not receive full credit. Please try to do all all work in the space provided
and circle your final answer.
1. (l6pts) Evaluate the following limits using L’Hôpital’s rule or any other method.
(a) (4pts) lim
x—O
(b) (4pts) lim
ex
—
1—
ln(x+1)
—
x—O
5jflX
-
4
(c) (4pts) lirn
L’14
xlnx
X
½)
+
2-EN
A4
—
—
—
1i
.j
I
-L-/--Th
l1.4’4
(d) (4pts) urn
cos (2x)
x—O
—
1
Il-I
—
12
lc’ (-x)
—
L/to (-x)
2. (8pts) Consider the sequence with first five terms given by
3
a
=
1
4
a
=
2
5
a
=
3
6
a4
7
a
5
Assuming the sequence continues on in this same manner, answer the following questions:
+
(a) (2pts) Find a formula for a. a =___________
(b) (2pts) lim a
=__________
c4Y
e
44’t
(c) (2pts) Is this sequence convergent or divergent? 7
S (write ‘diverges’ if the series diverges)
7
YC4
a =
(d) (2pts)
twc.
1
3. (l2pts) In this problem, you will compute the value of an improper integral in
two
steps.
(a) (8pts) Compute the definite integral
dx.
fx2e_x3
Note: Your answer should be a function of t.
f
3
—t
-
(Xe-
Jo
,t)c
-.
3
0
-e
(b) (4pts) Take the limit as t
—
cc of your answer in part (a) to find
I
I
I
‘Jo
fz.—X
Xe-
3
00
3
xe
I
lim
3
x2e_x
0
tc,oJ
I
Z)C.:
dz =
.1
-
(e
!/K4
dx.
)
4. (7pts) Use geometric series to write the repeating decimal .474747... as a fraction.
,7LL77L7
—
joc
—.
OgOOo
*7 oo
/
1
SeV-S
e.o w-’i”-
/oOo
-
i
4OD
7eTt’
10.c)
5. (8pts) Rewrite the following series as a collapsing series and compute its sum.
00
n=1
2—
b—
2 + 2n
Ai
i--—
—l.
L
v’t2-v
k
J- \
S
-
h r-)
o
/,y( -é’)
1-3:—
SN
2-
+
V
±
,.J+I
f’fr2-
2
1((A1
(
1fr2fsJ*(
f.I+?-)
‘—E:---
i7
6. (2Opts) Determine, by whatever method you wish, whether the following series are convergent or
divergent. Circle C’ if the series is convergent or ‘D’ if the series is divergent. You do not need to
show work.
D
>iln(n±3)
C
9n3±n2+1
D
ne
n=1
3
fl
+7
cos (2irri)
7. (lOpts) Find the first four terms of the power series representation
f (x)
=
=
x +...
3
x+c
1
x +c
2
0+c
c
=
co
=
3
c
2
c
1
c
What is the radius of convergence of this power series?
=
2-.
-
)
Ixf&/.
-
L1’
3
3
3x_3x
8. (9pts) Write an expression for a to give an example of the type of series indicated. There are many
possible answers for each blank.
Example:
a is a positive series when a
=
n.
n=1
(a)
a is an alternating series when a
n=1
9
a is an absolutely convergent alternating series when a
(b)
=
n=1
7
a is a conditionally convergent alternating series when a
(c)
=
n= 1
3
9. (lOpts) Determine the interval of convergence of the series
5n
n=1
‘fi 14e-%’V
4
C-O
s.
/0
I
kt/X2-I
I?(-2-)
c.
Ix-2-J
‘-I
so
-l
==3,
5-
oLt3:
X-3
DY—
co
4
&ve
(ç)fl
(—/) t-
•=)
rc.
1- i
x 7
I LAJ41-r vtI
4