Math 152
1. Determine whether or not the series converges.

n

a)
2

 n
n2
n
5 4
n

b)
2


e)
 n

 4n  5
n
n 1
1

d)
3
n
n2

 3n  1
3
 2
1

c)
n
n 1
3 2
 n
1 2
sin( 1 n )
n
n 1
2. For what values of p do the series converge?


ln n
a) 
n2
n
1
b) 
p
n
n2
p
ln n
3. Test each series for convergence. In a and b, also estimate the remainder.


n
a) 
n 1
b)
n
e

n2

ln n
n
c)
3 2

n2

n ln n
n
2
1
n ln n

d)
n2
n
2
 n
4. #28 10.3 Stewart

Find n so that
R n  10
for the series 
2
n2
1
n (ln n )
.
2
10.4
5. Does the series converge? Does the series converge absolutely or conditionally?


sin n
a) 
n
n 1
p 1
p
What
if p  1 ??
sin( n  2 )
b) 
n 1
n
6. Determine whether the series converges and if it does, estimate

a)
(  1)

n2
ln n

n
b)

Rn
.

n
(  1 ) sin( 1 n )
c)

n
(  1 ) ln( 1  1 n )
n 1
n 1
7.

a)

n2
n!
3  5  7  ...  ( 2 n  1 )

b)

n0
r

n
n!
c)
1 

1 

n 
n 1 

n
2
Find the radius and interval of convergence for each power series.

8.
n
(  1) ( x  1)

a)
n4
n 1

b)
n

n
(  1) ( x  1)
n4
n 1
n
2n
n
n
(  1) ( 3 x  1)
9. 

n
n
n 1

10. 
( 2 n )! ( x  2 )
( n! )
n0
n
but we will not be able to test the endpoints of the interval.
2
Find a power series about a=0 for each function. Give the radius and interval of
convergence.
11.
f ( x )  ln( 2  x )
12. 
x
1 x
3
f (0)  2
dx
13.
f ( x )  ln( 1  x
14.
f (x) 
2
)
2
ln( 1  x )
f(0)=0
x
x
2
15.
f (x) 
16.
f ( x )  x arctan
3x  2
x
Write as a series.
1
17. 
e
x
0
1 x
x
2
dx
.
1
18.  sin(
x
2
) dx
0
19. Write the Taylor series about
a 

for
f ( x )  sin x and g ( x )  cos x .
4
20. Find the Maclaurin series for
series for sin x.
f ( x )  sin x cos x
using an identity and the Maclaurin
5x
.
x
.
21. Find the Taylor series about a=1 for
f (x)  e
22. Find the Taylor series about a=3 for
f ( x )  xe
23. Find the Taylor polynomial of degree 4 about a=1 for f ( x ) 
error on [ 1, 1.2] if the polynomial is used to approximate f(x).
24. a) Find the Taylor polynomial of degree 3 about a=0 for
b) find the Taylor polynomial of degree 6 about a=2 for
25. Find the center and radius of the sphere x 2
P(2, 1, 3) inside or on or outside the sphere?
 y
2
 z
2
3
x  x
1 3
f ( x )  ln( 1  x )
. Estimate the
.
2
g ( x )  ln( 1  ( x  2 ) )
 8 x  10 y  16 z
.
. Is the point
26. Are the points P(1, 2, -1), Q(3, 0, 2) and R(2, 5, 7) collinear?
27. Find the angle between a=<1, -3, 7> and b=<2, 1, 5>.
28 and 29 are not on Exam 3 in Sp 2012
28. Find the area of the parallelogram with adjacent sides PQ and PR for P(2, 1, 7),
Q(3, -1, 5), and R(4, 2, 6).
29. Find the volume of the parallelepiped with the parallelogram of #21 as base and 3rd
adjacent side PS for S(1, 3, -1).
30. Find the 10th derivative at 2 of f ( x )  e ( x  2 ) .
2
31. Find the Maclaurin series for each.
32 .
f (x) 
x
3
2 2
(9  x )
2
33 .
f ( x )  ln( 16  x )
34 .
 x2
f ( x )  x arctan 
 4




