Homework 4 Solutions- Maclaurin and Taylor polynomials - Matthew Choi Assignment HW4 due at 11:59pm PST Math6B-01-W15-CHEN 1(1 pt Use the limit

Homework 4 Solutions- Maclaurin and Taylor polynomials -...

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Matthew Choi Math6B-01-W15-CHEN Assignment HW4 due 02/05/2015 at 11:59pm PST 1. (1 pt) Use the limit comparison test to determine whether n = 16 a n = n = 16 6 n 3 - 6 n 2 + 16 6 + 3 n 4 converges or diverges. (a) Choose a series n = 16 b n with terms of the form b n = 1 n p and apply the limit comparison test. Write your answer as a fully simplified fraction. For n 16, lim n a n b n = lim n (b) Evaluate the limit in the previous part. Enter as infinity and - as -infinity. If the limit does not exist, enter DNE. lim n a n b n = (c) By the limit comparison test, does the series converge, di- verge, or is the test inconclusive? ? Correct Answers: 6*nˆ4-6*nˆ3+16*n 3*nˆ4+6 2 Diverges 2. (1 pt) Each of the following statements is an attempt to show that a given series is convergent or divergent not using the Comparison Test (NOT the Limit Comparison Test.) For each statement, enter C (for ”correct”) if the argument is valid, or enter I (for ”incorrect”) if any part of the argument is flawed. (Note: if the conclusion is true but the argument that led to it was wrong, you must enter I.) 1. For all n > 1, 1 n ln ( n ) < 2 n , and the series 2 1 n diverges, so by the Comparison Test, the series 1 n ln ( n ) diverges. 2. For all n > 1, n 5 - n 3 < 1 n 2 , and the series 1 n 2 converges, so by the Comparison Test, the series n 5 - n 3 converges. 3. For all n > 2, 1 n 2 - 4 < 1 n 2 , and the series 1 n 2 converges, so by the Comparison Test, the series 1 n 2 - 4 converges. 4. For all n > 1, arctan ( n ) n 3 < π 2 n 3 , and the series π 2 1 n 3 con- verges, so by the Comparison Test, the series arctan ( n ) n 3 converges. 5. For all n > 2, ln ( n ) n > 1 n , and the series 1 n diverges, so by the Comparison Test, the series ln ( n ) n diverges. 6. For all n > 2, ln ( n ) n 2 > 1 n 2 , and the series 1 n 2 converges, so by the Comparison Test, the series ln ( n ) n 2 converges. Correct Answers: I I I C C I 3. (1 pt) Consider the series n = 1 1 n ( n + 3 ) Determine whether the series converges, and if it converges, determine its value.
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