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# exam1 - Test ∞ X n =123-1 n n 3 e n Problem 3(20 points a...

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Math 1502 B - McClain - Exam 1, 9/24/07 Name: Section/TA: PLEASE READ THIS FIRST: Write your name and section on the top of this page. Circle your ﬁnal answer. On the questions about inﬁnite series, state clearly which test you are using. The test has FIVE problems. First solve the problems you can do immediately; postpone the others until later. The total possible is 109 points, however, you will be scored out of 100. GOOD LUCK!

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Problem 1. (20 points) C a) (16 points) Find the general solution. xy 0 + 2 y = cos ( x ) x b) (4 points) Find the particular solution, using the initial value condition y(1)=0.
Problem 2. (24 points) a) (7 points) Does the series converge? X k =3 1 k (ln k ) 2 b) (7 points) Does the series converge? X k =1000 3 k 2 + 5000 k k 6 - 800 k 5 + 3 k

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c) (10 points) Test the following series for convergence using the Alternating Series

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Unformatted text preview: Test. ∞ X n =123 (-1) n n 3 e n Problem 3. (20 points) a) Find the interval of convergence of the following series: ∞ X k =1 kx k 2 k Problem 4. (20 points) We want to approximate e 1 3 using Taylor polynomials. a) (12 points) Using the remainder bound, ﬁnd an n so that the error is at most .01 if we use the n th Taylor polynomial P n ( 1 3 ) for the approximation. b) (8 points) Using the correct n from part (a), approximate e 1 3 . Problem 5. (25 points) a) (20 points) Sum the series ∑ ∞ n =1 n 2 n (Hint: consider the series for f ( x ) = 1 1-x ). b) (5 points) What is the 22 derivative of f ( x ) evaluated at zero? (i.e. f (22) (0))...
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exam1 - Test ∞ X n =123-1 n n 3 e n Problem 3(20 points a...

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