1220-90 Exam 2
Summer 2013
KEY
Name
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
arid circle your final answer.
1. (l6pts) Evaluate the following limits using L’Hôpital’s rule or any other method.
(a) (4pts) thn
--
(c) (4pts) lim x ln x
CO.S )C I(Sik’.X
ILM
cos x
(b) (4pts) lim
x—O tan (2x)
i
L2-
Hint: Rewrite as a fraction of indeterminate type
1i
—
—
O
?(401
(d)
(4pts)
lirn
sinx—x
1.
It’M
-)/)L
LoC
—
I
—
3ç2
—
)(‘—70
2. (8pts) Consider the sequence with first five terms given by
=1
1
a
2
a
=
2
4
a
=
3
8
a4=
16
a
5
Assuming the sequence continues on in this same rnai ier—aiswer the following questions:
/7_I
(a) (2pts) Find a formula for a. a
=
( /!
h-l
(b) (2pts) lim a
(c) (2pts) Is this sequence convergent or divergent?
(d) (2pts)
a
9-
3
(write ‘diverges’ if the series diverges)
w/
)
4
-l)lr---
CDb44rfD
3. (llpts) In this problem. you will compute the value of an improper integral in two steps.
(a) (7pts) Use integration by parts to compute the definite integral
dx.
Note: Your answer should be a function of t.
7
I
‘Jo
(—?e
€-
t.
J,
t
-)c
t
oLA
otA.LpC
i-(’-e
i
v
-i-f
(b) (4pts) Take the limit as t
—
no of your answer in part (a) to find
ft
I
xe_S
Jo
dx
5 dx.
= km I xe
0
t—*o0J
fhM
=1.
4. (l5pts) All of the following series are convergent. Evaluate them.
00
(a) (5pts) 2
()
n—i
C.
.cee-
2.
2-
cc
/
.—
1
1
cos (nit)
(b) (5pts)
—
-
3n_1
3
I
3’-’
n= 1
_).,
r-
4—
-i
00
(c) (5pts)
::
n=i
n+1
—(V)
KE
3
n+2
‘4
b%t
-
=6
4
A
—3
(3
—3
2
—
2-
L
13
)•
5. (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.
1
D
n=1
-
D
2
42 + 9
c:n
n=1
1
D
n=1
(—1)
D
lnn
n=1
n2
TLl
c
G,
n1nn
n=1
C
n=1
C®
6. (lOpts) Find the first four terms of the power series representation
f (x)
I
co=
=
1
(1-x/2)-
c=
=
x +...
3
0 +c
c
x + c
2
x+c
1
I
b
c
/
3
=
2
C3
10
What is the radius of convergence of this power series?
-
-i
()
D
I
/2-i
()
iii.
t4-/x(i,
J
3
L
+
33
1)
°)xI2.
..
=
C)
—
C)
.
—
C
—
Cfl
1E
-
+
4—
•
—
—-‘i
i
--
—
p
I
C)
C)
C)
C
C)
C)
C
C
C)
C
C)
C)
C)
4—
C
C
C)
C)
4--
(‘C
4—
c)
x
U
I
9)
t
4
I
c_I’
0
I