Math 152 In Class Exam 3 Review
Find the sum or show the series is not convergent.

1
1. 
[

n
n
10

 1

1

1
1 
ln ln
2. 
 n

n1


]
n2
3.
n
1

1




ln
n ln(
n2
)

n

2


1
1 

cos
cos


n
n
1


n

1
4. 

5. 
n sin


 1
3

 2
n 1
6. 
n
n 1

4
2

7.

n
n
(
n

1
)
3 

n
1

1

8. 
(  1)
3
n 5
n
n
Test for convergence.

n
9. 
n 1
e
Also estimate the remainder,
n

10. 
n! e
Rn
.
n
n 1

3
5
7
...(
2
n
1
)
11. 
n
n
!
2
n

0

10
12. 
n  0
n
What is the sum?
n!
Test for convergence and estimate

13. 
(  1)
n2
n
ln n

14. (1)
n
n1
15. (1)
n
1
tan
n
1
nsin
n
Rn
for each convergent series.
n
16. . Test for convergence. State the test or tests you use.

a)

sin
2
n  2 cos
n
n 1

n 

cos

2

n 1
 3 n  12 )
2

c)




1

n3
n
1 .2

b)
2
n(
n  ln n )

3
17. Estimate R 5 for  n e
n
2
n 1
Find a power series and give the radius and interval of convergence for each function.
18.
f (x)  arctan
x
2
19.
g(x) xarctan
x
20.
f (x) ln(
4 x)
21.
g(x) ln(
4x )
22.
23.
2
x
f (x) 
2
(1 x)
1
f (x) 
2
(52x)
Find the Taylor series about a and find the radius and interval of convergence for each
function.
24.
f ( x) 
x
25.
g(x) 
x 2
, a=4
, a=2
26.
f ( x)  sin x
, a=0
27. f(x) = cos x , a=0
28.
29.
30.
f ( x)  e
x
, a=0
x
g( x)  xe
2
, a=0
sinx

 x
f ( x)  


 1
x 0
x 0
Find the nth degree Taylor polynomial about a and estimate
R
)
| f(x
)
T
)|.
n(x
n(x
29. f(x) = tan x , a = 0, n = 3 |x|<π/6

30. f(x) = tan x , a =
2
31.
g(x) = tan (x
),
32.
h(x) = tan (x  2 )
4
, n=3
| x   / 4 | 0 . 1
a = 0, n = 6
2
, a = 2, n=6
33. f(x) = sec x a=0, n=2
34. f(x)=sec x a=π/4, n=2
35. Find the distance between the centers of the two spheres,
2
2
2
2
2
2
x  y  z  6 x  4 y  8 z  7 and x  y  z  2 x  8 y 
4z  4.
Do they intersect?