Name
ID
MATH 172
EXAM 3
Fall 1998
Section 502
a.
b.
c.
d.
e.
/50
11
/15
12
/15
13
/10
14
/15
P. Yasskin
Multiple Choice: (5 points each)
1. Compute
1-10
3n 2
lim
nv. 1 n 3
0
1
2
3
Divergent
2. Find
r
such that
5 5r 5r 2 5r 3 5r 4 a. 2
5
C 3.
b. " 2
c. 3
5
d. 5
3
5
e. " 2
3
!
.
3. The series
n1
1
n2 n
is
a. divergent by comparison to
!
!
.
n1
b. convergent by comparison to
1 .
n
.
n1
1 .
n2
c. divergent by the ratio test.
d. convergent by the ratio test.
e. divergent by the n th -term test.
1
!
99
4. Compute
1
k
k1
a.
b.
c.
d.
e.
"
1
k1
.9
.99
1
1.1
Divergent
!
.
5. Compute
3n 2
1 n3
n1
a. ln 2
b. 3
2
c. 27
82
d. Convergent but none of the above
e. Divergent
!Ÿ
.
6. The series
n1
a.
b.
c.
d.
e.
"1
n
3 n
is
absolutely convergent.
conditionally convergent.
absolutely divergent.
conditionally divergent.
oscillatory divergent.
7. Compute
lim
xv0
Ÿ cos 2x " 1 2x 2
x4
a. 0
1
24
c. 1
12
d. 2
3
b.
e. .
2
!
.
8. Given that
a.
b.
c.
d.
e.
xn n0
1
1"x
1
1"x
x
1"x
x
1"x
n
1"x
Ÿ
Ÿ
1
1"x
(for |x| 1),
then (for |x| 1) we have
!
.
n xn n0
2
2
!
.
9. The series
n0
1
n!
1
2
n
converges to
a. ln 2
b. e
c. sin 1
Ÿ 2
d. sin 2
e. e 2
10. Find the
3 rd degree
x 2.
a. 3 x " 2 3
8
b. "3 x " 2 3
8
c. "6 x " 2 3
d. "1 x " 2 3
16
e. 1 x " 2 3
16
Ÿ
Ÿ
Ÿ
term in the Taylor series for
Ÿ f x 1x
centered at
Ÿ Ÿ 3
!Ÿ
.
11. (15 points) Find the interval of convergence for the series
n0
x"5 n
.
3nn3
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:
4
Ÿ 12. (15 points) Let
f x x 2 cos x.
a. (10 pt) Find the Maclaurin series for
and also write out the first 4 terms.
b. (5 pt) Find
Ÿ fx .
Write the series in summation form
Ÿ f Ÿ6 0 .
5
C
Ÿ
Ÿ find the 6 th degree Taylor
ln 1 t t " 1 t 2 1 t 3 " ,
2
3
polynomial approximation about
for
x0
ln 1 x 2 .
13. (10 points) Given that
14. (15 points) You are given:
e
;
"x 2
!Ÿ
.
n0
"1
n
x 2n 1 " x 2 x 4 " x 6 2
6
n!
a. (10 pt) Use the quadratic Taylor polynomial approximation about
e
"x 2
to estimate
0.1
e
"x 2
dx.
x0
for
(Keep 8 digits.)
0
b. (5 pt Extra Credit) Your result in (a) is equal to
much? Why?
C.
;
0.1
e
0
"x 2
dx
to within o how
6