MATH 270 - Week 4 iLab 2. Questions with Answers. A+ Guide.

Math 270 Name: Lab #4

The Taylor and Fourier Series are among the most important ideas in mathematics that you will
encounter during this class. Remember to SHOW ALL YOUR WORK TO EARN FULL CREDIT,
because providing only the answers will earn approximately 1/3 of the available credit.
Concepts include: Operations and Computations with Maclaurin, Taylor, and Fourier Series.
n
 1
1.) Find the first 5 terms of the sequence: an  4    (n  1)! , n = 0,1,2,3,…
 2
0
 1
n=0: a0  4    (0  1)!  4  1  3
 2
1
 1
n=1: a1  4    (1  1)!  2  2  4
 2
2
 1
n=2: a2  4    (2  1)!  1  6  5
 2
3
 1 1 49
n=3: a3  4    (3  1)!   24 
 2 2 2
4
 1 1 479
n=4: a 4  4    (4  1)!    120 
 2 4 4

Final Answer:
49 479
a0  3, a1  4, a2  5, a3  , a0 
2 4

2
2.) Find the sum of the finite geometric series:  2(2n  1)
n 0
n1
Show all work for full credit.

n=0: 2(2  0   1)  2
0 1


n=1: 2(2  1  1)  2
11


n=2: 2(2  2   1)  54
21


Sum: S  2  2  54  54

Final Answer:
2

 2(2n  1)
n 0
n 1
 54

 n
3.) Give the first four terms of the infinite series for an = sin   , n  0,1,2,... . Write the
 4
infinite series in sigma notation.

  0  
n=0: a0  sin    sin  0   0
 4 
  1    2
n=1: a1  sin    sin   
 4   4 2




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,Math 270 Name: Lab #4

  2    
n=2: a2  sin    sin    1
 4   2
  3    3 2
n=3: a3  sin    sin   
 4   4 2
n

2 
Sigma notation:  
n 1 
 2


Final Answer:
2 2
a0  0, a1  , a 2  1, a3 
2 2
n
  2

n 1 

2


50

 2.45  0.77 
n1
4.) Find the sum of the finite geometric series using your CAS SYSTEM:
n 5




Final Answer:
50

 2.45  0.77 
n 1
 2.22014
n 5




  (n 1)
3  8
5.) Find the sum of the finite geometric series using your CAS SYSTEM:   
n 0 5  7




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, Math 270 Name: Lab #4




Final Answer:
  (n 1)
3 8 21

n 0 5 
 7

5

6.) Find the first six partial sums of the following series to determine if it converges
n

 1
or diverges.  2 
n 0  3


0 1
 1   1 2 4
S1  2    2   2  
 3  3 3 3
2
2  1 2 2 14
S 2   2   2    2    
3  3 3 9 9
3
2 2  1 2 2 2 40
S 3  2    2    2    
3 9  3 3 9 27 27
4
2 2 2  1 2 2 2 2 122
S 4  2     2    2     
3 9 27  3 3 9 27 81 81
5
2 2 2 2  1 2 2 2 2 2 364
S5  2      2    2      
3 9 27 81  3 3 9 27 81 243 243
6
2 2 2 2 2  1 2 2 2 2 2 2 1094
S 6  2       2    2       
3 9 27 81 243  3 3 9 27 81 243 729 729

Convergent or divergent? Convergent

Final Answer:


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