CALCULUS BC
REVIEW SHEET 2 ON SERIES
Work the following on notebook paper. Do not use your calculator.
1. Which of the following is a term in the Taylor series about x = 0 for the function f  x   cos  2 x  ?
(A) 
1
(B) 
x2
4
x3
(C)
2
x4
(D)
1
x5
(E)
4
x6
3
60
2
3
45
___________________________________________________________________________________________
 x  2
 n 3 n
n 1
 
n

2. Find the values of x for which the series
converges.
(A) x = 2
(B) 1  x  5
(C) 1  x  5
(D) 1  x  5
(E) All real numbers
___________________________________________________________________________________________
x5
x7
x 2n1
3. The series x  x3 

 ... 
 ... is the Maclaurin series for
2!
3!
n!

(A) x ln 1  x 2


(B) x ln 1  x 2

(C) e x
2
(D) xe x
2
2
(E) x2e x
___________________________________________________________________________________________
4. The coefficient of x 3 in the Taylor series for e 2x at x = 0 is
1
1
2
4
8
(A)
(B)
(C)
(D)
(E)
6
3
3
3
3
___________________________________________________________________________________________

3n
 2 
5. Let f  x     cos x  . Evaluate f 
.
 3 
n 1
6. Find the sum of the geometric series
9
8

3
4

1
2

1
3
 ...
7. Find the Taylor polynomial of order 3 at x = 0 for f  x   1  x .
___________________________________________________________________________________________
8. The function f has a Taylor series about x = 2 that converges to f  x  for all x in the interval of
 n  1 !
n
convergence. The nth derivative of f at x = 2 is given by f    2  
for n  1, and f  2   1.
3n
(a) Write the first four terms and the general term of the Taylor series for f about x = 2.
(b) Find the radius of convergence for the Taylor series for f about x = 2. Show the work that leads to your
answer.
(c) Let g be a function satisfying g  2   3 and g   x   f  x  for all x. Write the first four terms and the
general term of the Taylor series for g about x = 2.
(d) Does the Taylor series for g as defined in part (c) converge at x  2? Give a reason for your answer.
___________________________________________________________________________________________
3 

9. Let f be the function given by f  x   cos  3 x 
 and let P  x  be the third-degree Taylor
4 

polynomial for f about x = 0.
(a) Find P  x  .
(b) Use the Lagrange error bound to show that
1
1
1
f    P  
.
6
 6  300
(c) Let G  x  be the function given by G  x    f  t  dt . Write the third-degree Taylor polynomial, T  x  ,
x
0
for G about x = 0.
Answers to Review Sheet 2 on Series
1. C
2. C
3. D
5. 
1
6.
9
27
4. D
7. 1 
40
x
2

x
2
8

x
3
16
2  x  2 3 x  2
4  x  2
 n  1 x  2   ...
8. (a) 1 



...

3
32
33
3n
2
3
n
(b) Radius = 3
 x  2
(c) 1  x 
3
2
 x  2

9
3
 x  2

27
4
 x  2
 ... 
3n
n 1
 ...
(d) No, x  2 is outside the interval of convergence, which is 1  x  5.
9. (a) P  x  
2 3 2 x 9 2 x 2 27 2 x3



2
2
2  2!
2  3!
(b) R3  x  
(c) 
f  4  z  x  0 
4
4!
2
3 2 2 3 2 3
x
x 
x
2
4
4
4
81x 4
1
1
1
 1   81   1 

so R3         


4
4!
4!2
 24 16  300
 6   4!   6 