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AP Calculus BC
Unit 9: Taylor and Maclaurin Series
Take-home test
Multiple Choice: Show all work in order to receive full credit.

1. What are all the values of p for which
x
1
2p
dx converges?
1
a.
b.
c.
d.
e.
p < -1
p>0
p>½

p>1
There are no values of p for which this integral converges.

2. What are all values of x for which the series

x  2n
n 1
a.
b.
c.
d.
e.
-3 < x < -1
-3 ≤ x < -1
-3 ≤ x ≤ -1
-1 ≤ x < 1
-1 ≤ x ≤ 1
n
converges?


3. What is the value of
a.
b.
c.
d.
e.
2 n 1
 3n ?
n 1
1
2
4
6 
The series diverges.

4. What is the radius of convergence of

n  1
a.
b.
c.
d.
e.
3
2
1
0

(1) n
 1
( x  2) n
?
n  3n
5. Which of the following series diverges?

I.
 sin 2 




n 0  

II.

n 1
n
1
n
3
 en 

 n 
n 1  e  1 

III.
a.
b.
c.
d.
e.
III only
I and II only
I and III only
II and III only
I, II, and III
6. What are the first four non-zero terms in the power expansion of e 4 x about x = 0?
x 2 x3
1

x


a.
2 3
b. 1  4 x  8 x 2 32 x 3
2
c. 1  4 x  2 x 2  x3
3
32
d. 1  4 x  8 x 2  x 3
3
64
e. 1  4 x  8 x 2  x 3
3
7. What are all the values of x for which the series x 
a.
b.
c.
d.
e.
1  x  1
1  x  1
1  x  1
1  x  1
All real x
x2 x3 x4
   ... converges?
2 3 4
 1  

 2n  !
n 0

8. Evaluate the sum (hint: figure out which power series this resembles):
n
2n
a. 1
b. -1
c. π

d.
2
e. e

9. Let f  x  be a function defined by the power series f  x    2 x n . If g '  x   f  x 
n 0
and g  0  2 , then g  x  

a. 2   2 x n
n 1

b. 2 nx n 1
n 1

2 xn
n 1 n

2 n1
d. 
x
n 0 n  1

2 xn
e. 4  
n 1 n
c. 2  
10. The Taylor series expansion of

a.

 1  x  2 
n
2n 1
n 0

b.

 1  x  2 
n 1
2 n 1
n0

c.

n0

d.

n0

e.

n 0
n
 x  2
n
2n 1
 x  2
n
2n
 1  x  2 
n
2n
n
n
1
about the point a  2 is
x

11. Find the number of terms necessary to approximate the sum of the series


n 1
1
3
n2
with an error of less than 0.001.
a. 10 2
b. 4,000,000
c. 1,000
d. 4,000
e. None of these
12. Use the third-degree Maclaurin polynomial to approximate the value of e0.2 . Round your
answer to four decimal places.
a. 1.2213
b. 1.2214
c. 1.2227
d. 2.6667
e. None of these
13. What is the coefficient of x3 in the Taylor series for e2x at x = 0?
1
a.
6
1
b.
3
2
c.
3
4
d.
3
8
e.
3

14. What is the sum of the Maclaurin series
3
5
7
 2n  1
n
 


   (1)
 
3!
5!
7!
(2n  1)!
a. 1
b. 0
c. -1
d. e
e. There is no sum
Short Answer (show your work):
1. Find the fourth-degree Taylor polynomial for f(x) = ln x, centered at c = 1.

2. Consider the series

n  1
(1) n  1 ( x  1) n
n
.
a. Find the value of x at which the series is centered.
b. Find the radius of convergence.
c. Find the interval of convergence.
3. If f ( x) 


n  1
(1) n ( x  2) n
, find the interval of convergence of
n

f ( x) dx .