Answer the following questions showing all your steps to earn full credit.

 

 

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 2.) Find the sum of the finite geometric series: 2(2n 1) n1 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.) Name __________________________ Lab #4 Find the sum of the finite geometric series using your CAS SYSTEM:
50 2.45 0.77 n1 n 5 5.) Find the sum of the finite geometric series using your CAS SYSTEM: (n 1) 38 n 0 5 7 6.) Find the first six partial sums of the following series to determine if it converges
n 1 or diverges. 2 3 n 0
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. Page 2 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) = e 2x 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 = ______________________ Page 3 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
x2 x3 x 4
1 x 2! 3! 4!
x 3 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(3x 5). 13.) Use one of the expansions above to write the Maclaurin Series for (3 + x) 4 Page 4 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 sin x 2 dx . Show the ‘Maclaurin integral’, the integral of the Maclaurin expansion of the 0 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)
Page 5 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)? Page 6 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) = ________________________________________________________________
_______________________________________________________________
_______________________________________________________________ Page 7 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) 0x
3x
Find each value correct to four decimal places. a0 = ____________________________________________________________ a1 = ____________________________________________________________ a2 = ____________________________________________________________ a3 = ____________________________________________________________ b1 = ____________________________________________________________ b2 = ____________________________________________________________ b3 = ____________________________________________________________ f(x) = ________________________________________________________________
_______________________________________________________________
_______________________________________________________________ Page 8

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