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Math 270
Name __________________________
Lab #4
SHOW ALL YOUR WORK TO EARN FULL CREDIT, because providing only the answers will
earn approximately 1/3 of the available credit.
n
1.)
 1 
Find the first 5 terms of the sequence: an  4    (n  1)! , n = 0,1,2,3,…
 2
2.)
Find the sum of the finite geometric series:
2
 2(2n  1)
n1
Show all work for full credit.
n 0
3.)
Give the first four terms of the infinite series for
the infinite series in sigma notation.
Page 1
an =
 n 
sin   , n  0,1,2,... . Write
 4 
Math 270
4.)
Find
Name __________________________
the
sum of
50
 2.45  0.77
the finite
geometric series
Lab #4
using your
CAS
SYSTEM:
n1
n 5

5.) Find the sum of the finite geometric series using your CAS SYSTEM:
38

 
n 0 5  7 
 (n 1)
6.) Find the first six partial sums of the following series to determine if it converges
 1 
or diverges.  2  
 3 
n0

n
S1 = _______________________
S2 = _______________________
S3 = _______________________
S4 = _______________________
S5 = _______________________
S6 = _______________________
Convergent or divergent?
7.)
Find the first six partial sums of the following series to determine if it converges
2 3 4
5
6


 ...
or diverges. 1   
3 9 27 81 243
S0 = _______________________
S1 = _______________________
S2 = _______________________
S3 = _______________________
S4 = _______________________
S5 = _______________________
Convergent or divergent?
Circle one.
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Math 270
Name __________________________
Lab #4
8.) When writing a transcendental equation as a polynomial, when would a Taylor series be
a better choice than a Maclaurin series to approximate a function ?
9.) Determine the Maclaurin Series for the following function: f(x) = e2x for n = 4
10.) Enter f(x) = e2x and the series approximation from the problem above into your CAS
SYSTEM.
Sketch the graphs. Approximate the values of x for which the Maclaurin series appear
to converge on f(x) = e2x
Interval of convergence = ______________________
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Math 270
Name __________________________
Lab #4
11.) Use your CAS SYSTEM to find a Maclaurin Series approximation for f(x) = x sin(2x)
where n = 6.
Function:
e
sin(x) )
cos(x)
ln(x  1)
(x  1)n
Maclaurin Series expansions
Series Expansion
x 2 x3 x 4
1 x 


2! 3! 4!
x3 x5
x

3! 5!
x2 x4
1

2! 4!
x2 x3 x 4
x


2! 3! 4!
n(n  1) 2 n(n  1)(n  2) 3 n(n  1)(n  2)(n  3) 4
1  nx 
x 
x 
x ...
2!
3!
4!
For problems 12–16, use the expansions above.
12.) Use one of the expansions above to write the Maclaurin Series for sin(3x5).
13.) Use one of the expansions above to write the Maclaurin Series for (3 + x)4
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Math 270
Name __________________________
Lab #4
14.) Use one or more of the expansions above to write the first 3 terms of the Maclaurin
Series for x2e2x.
15.) Use four terms of a Maclaurin series to approximate the value of the following:
1
 sinx
2
dx . Show the ‘Maclaurin integral’, the integral of the Maclaurin expansion of
0
the function.
16.) Calculate each value below using five terms of a Maclaurin series expansion. Then find
the calculator value correct to 8 places.
e0.2
cos(3.57)
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Math 270
Name __________________________
Lab #4
ln(2.8)
17.) Graph ln(x), the Maclaurin series used in problem 16 and then the Taylor series found in
problem 17. Change your viewing window (green diamond F2) to xmin = -.5,
xmax = 10, xscl =.5, ymin= -3, ymax = 5 , yscl = 1. Then sketch and label each graph.
Which series is a closer fit to ln(x) ? Why? For what values of x can these series be used to
approximate f(x)?
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Math 270
Name __________________________
Lab #4
18.) Use your calculator to determine a Taylor Series approximation for f(x) = cos (x) where
n = 4 and a = 3.
19.) Find the Fourier series for the function f(x) = sin x , for 0 < x < 2
Give the values correct to four decimal places.
a0 = _________________________
a1 = _________________________
a2 = _________________________
a3 = _________________________
b1 = _________________________
b2 = _________________________
b3 = _________________________
f(x) = ________________________________________________________________
_______________________________________________________________
_______________________________________________________________
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Math 270
Name __________________________
Lab #4
20) Find at least three non-zero terms and at least two cosine and two sine terms of the
2x    x  0
Fourier series for f(x)  
0x
3x
Find each value correct to four decimal places.
a0 = ____________________________________________________________
a1 = ____________________________________________________________
a2 = ____________________________________________________________
a3 = ____________________________________________________________
b1 = ____________________________________________________________
b2 = ____________________________________________________________
b3 = ____________________________________________________________
f(x) = ________________________________________________________________
_______________________________________________________________
_______________________________________________________________
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