Calculus BC
Wks 9-F
Name _____________________
Date ______________________
On problems 1 and 2, find the radius of convergence and the interval of convergence.

1.

n 1
 1n x  5n

2.
4n n
3. 1996 BC2
x 2n

n 0 2n !
The Maclaurin series for f  x  is given by 1 
x x 2 x3
xn
    
 
2! 3! 4!
 n  1!
(b) For what values of x does the given series converge? Show your reasoning.
(c) Let g  x   x f  x  . Write the Maclaurin series for g  x  , showing the first three nonzero
terms and the general term.
(d) Write g  x  in terms of a familiar function without using series. Then write f  x  in terms
of the same familiar function.
4. 1994 BC5 Calculator Allowed
2
Let f be the function given by f  x   e 2 x .
(a) Find the first four terms and the general term of the power series for f  x  about x  0.
(b) Find the interval of convergence of the power series for f  x  about x  0. Show the analysis
that leads to your conclusion.
(c) Let g be the function given by the sum of the first four nonzero terms of the power series for
f  x  about x  0. Show that f  x   g  x   0.02 for 0.6  x  0.6 .
5. 1991 BC5
Let f be the function given by f  t  
4
and let G be the function given by
1 t2
G  x    f  t  dt .
x
0
(a) Find the first four nonzero terms and the general term for the power series expansion of
f  t  about t  0.
(b) Find the first four nonzero terms and the general term for the power series expansion of
G  x  about t  0.
(c) Find the interval of convergence of the power series in part (b). (Your solution must include
an analysis that justifies your answer.)
6. 2003 BC6 The function f has a Taylor series about x = 2 that converges to f  x  for all x in the
interval of convergence. The nth derivative of f at x = 2 is given by f  n   2  
n  1, and f  2   1.
 n  1 !
3n
for
(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.
Selected Answers:
1. 4; (1,9]
2.  ; converges for all x
3. 1996 BC2
b.
x 2 x3
x n 1
 
c. g  x   x     
2! 3!
 n  1!
  ,  
d. g  x   e  1,
x
 ex 1
 x ,
f  x  
 1,

x0
x0
4. 1994 BC5
 1 2n x 2n  
4 x4 8x6
a. e
 1  2x 

  
2!
3!
n!
5. 1991 BC5
n
a. f  t   4  4t 2  4t 4  4t 6    4  1 t 2 n  
n
2 x 2
2
b.
  ,  
8
c.
4  1 x 2 n 1
4 x3 4 x5 4 x 7
b. G  x   4 x 


  
 
3
5
7
2n  1
c.  1  x  1
6. 2003 BC 6
n
2  x  2 3 x  2
4  x  2
 n  1 x  2   ...


 ... 
(a) 1 
2
3
3
3
3
3n
(b) Radius = 3
2
3
4
n 1
x  2
x  2
x  2
x  2






 ... 
 ...
c 1  x 
3
9
27
3n
2
or 3   x  2  
 x  2
3
2

 x  2
n
3

 x  2
4
 ... 
 x  2
16 .6 
 .011  .02
4!
n 1
 ...
3
9
27
3n
(d) No, x  2 is outside the interval of convergence, which is 1  x  5.