Exam 3 Review Math 152
1. Test each for convergence and find the sum when possible.

1
n 1
n
a )  n sin

b) 
n 1
2

4
4 
c)  
 n
5 
n 1  n ( n  1)

3
n
 (  1) n  2 n  1 

  2 n 1

( 2 n  1)! 
n  0 3

d)
2. Test for convergence.

n
a) 
e
n 1
also estimate the remainder
n

b )  n! e
n
n 1

c)  n r
2
n
| r | 1
n 1
3  5  7 ...  ( 2 n  1)

d) 
n!2
n 1

e) 
n0
e
n
Only show the ratio test is inconclusi ve.
n
n!
3. Test for absolute and conditional convergence and estimate the remainder when convergent.
(  1)

a) 
n
ln n
n2

b )  (  1) tan
n
n 1

c )  (  1)
n

1
n
n 1 
n

n 1
4. For which series is the ratio test inconclusive? Which test works?

a)

n2

n ln n
2
b)
n

n2
ln n
n
2

c)

n 1
( n  1)
n!
3
 n2 1
  4
3
n  3 n  2 n

d)
Find a power series about a=0 .Find the radius and interval of convergence for each.
5. f(x) = arctan x
6. g(x) = x arctan x 2
7.
f (x) ln(
4x)
2
8.
g
(x)ln(
4x )
9.
x
f (x) 
2
(1x)




10.
1
f (x)
2
(52x)
Find the Taylor series about a and find the radius and interval of convergence for each.
11. f ( x )  ln x
a  2
12. f ( x ) 
a 1
13.
x
f ( x )  xe
x
a  2
14. Find the tenth derivative at 0, f
(10 )
2
( 0 ) , for f ( x )  x cos x .
Find the Maclaurin series for each.
15. a ) f ( x )  cos x
b ) f ( x )  sin x
 sin x

x  0
 1
x  0
16. f ( x )   x
17. g ( x )  xe  x
c) f ( x)  e
x
2
Find the nth degree Taylor polynomial about a and estimate the error , | f ( x )  T n ( x ) | on the given
interval.
18. f ( x )  sin x
a 
19. f ( x )  1  x
20. f ( x )  1  x 2

n 3
|x
6
a  0

|
6
n  2
| x |
6
1
4
a  0
n  4

| x |
1
2
Sections 11.1, 11.2
21. Find the center and radius of x 2  y 2  z 2  4 x  6 y  7 z . Is the center inside or outside the
sphere x 2  y 2  z 2  4 x  4 z ?
22. Find the intersection of the graph of f ( x , y )  3  x 2  y 2 in each plane.
a) x-y plane
b) x-z plane
c) y-z plane
d) the plane x = 1
23.

b   2 ,3 , 4

and
a  3 ,  4 ,5 find

i)
comp

a
b

ii)
proj

a
b
iii) cos 
for   the angle between


a and b .