Name
ID
MATH 172
EXAM 3
Section 504
Spring 1999
1-10
/60
11
/25
12
/15
P. Yasskin
Multiple Choice: (6 points each)
1. Compute
a.
b.
c.
d.
e.
".
"1
lnn
lim
nv. n
0
e
.
!
.
2. The series
n1
a.
b.
c.
d.
e.
lnn
n
is
Divergent by the n th Term Divergence Test
Divergent by the Integral Test
Convergent by the Integral Test
Divergent by the Ratio Test
Convergent by the Ratio Test
3. A convergent sequence is recursively defined by
lim a n .
Find
a. 19
a1 1
and
a n1 3 " an
.
2 an
nv.
24
21 " 3
b.
2
c.
=
4
3
d.
2
e. 2
3
1
!Ÿ
.
4. The sum of the series
"1
n1
n1 n1
3
4n
is:
a. nonexistant since its partial sums oscillate
b. " 3
c. 9
4
7
d. nonexistant by the n th Term Divergence Test
e.
3
16
!Ÿ
.
5. Compute
a. 9
k2
Ÿ
3
. (HINTS: partial fractions,telescoping sum)
n"1 n1
4
b. 3
2
c. 1
d. 1
2
e. .
!Ÿ
.
6. The series
"1
n1
n
3nn2
n!
is
a. Absolutely convergent
b. Conditionally convergent
c. Divergent
!Ÿ
.
7. The series
n1
"1
Ÿn3 3 2
n
n100
2n
is
a. Absolutely convergent
b. Conditionally convergent
c. Divergent
2
Ÿ
2
3
4
ln 1 x x " x x " x 2
3
4
ln 1 2x " 2x 2x 2
lim
xv0
2x 3
a. 0
b. 8
3
c. 4
3
d. 1
3
8. Given that
Ÿ
C,
compute
Ÿ e. .
!Ÿ
.
9. For what values of x does the series
a.
b.
c.
d.
e.
x
x
0
x
x
n0
4 only
3 only
x 3 only
1 or x 3 only
2 or x 4 only
10. In the Maclaurin series for
Ÿ sin x 2
x
1
x"3
n
converge?
the coefficient of
x9
is
a. 1
1
3!
c. 1
5!
d. 1
7!
e. 1
9!
b.
3
11. (25 points) You are given:
Ÿ !Ÿ
f
cos x 2 .
"1
Ÿ2nx ! 1 " x2! x4! " x6! C.
n
a. (5 pt) If
f x cos x 2 ,
Ÿ Ÿ find
n0
Ÿ6 b. (5 pt) If
f x cos x 2 ,
Ÿ Ÿ find
f Ÿ16 0 .
;
4n
4
8
12
Ÿ0 .
Ÿ c. (10 pt) Use the quartic (degree 4) Taylor polynomial approximation about
for
Ÿ cos x
2
to estimate
0.1
d. (5 pt) Your result in (c) is equal to
Why?
Ÿ dx.
cos x
0
;
0.1
0
x0
2
Ÿ cos x 2 dx
to within o how much?
4
!Ÿ
.
12. (15 points) Find the interval of convergence for the series
n0
x"6 n
.
2n n
Be sure to identify each of the following and give reasons:
(1 pt) Center of Convergence:
a
Radius of Convergence:
(1 pt) Right Endpoint:
R
(5 pt)
x
At the Right Endpoint the Series
Converges
(circle one)
(3 pt)
(circle one)
(3 pt)
Diverges
(1 pt) Left Endpoint:
x
At the Left Endpoint the Series
Converges
Diverges
(1 pt) Interval of Convergence:
5