fall17mth143.practice2.3-ConvergenceTestsI.pdf

Thus the integral and the sum converge b again we

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Thus the integral, and the sum, converge. B. Again, we note that a n = n + 3 n 2 + 6 n + 3 is a sequence of positive decreasing terms, so that the Integral Test applies. We compare n = 1 n + 3 n 2 + 6 n + 3 with R c n + 3 n 2 + 6 n + 3 dn . Taking c = 1, we have Z 1 n + 3 n 2 + 6 n + 3 dn = lim n 1 2 ln ( b 2 + 6 b + 3 ) - 1 2 ln ( 10 ) . Because lim n ln ( b 2 + 6 b + 3 ) diverges, we know that the integral, and thus the sum, diverge. Answer(s) submitted: (incorrect) Correct Answers: 1/6ˆn 1/(6*ln(6)) A (n+3)/(nˆ2 + 6 n + 3) diverges B 4. (1 point) For each of the following series, indicate whether the integral test can be used to determine its convergence or not, and if not, why. A. n = 1 ln ( 0 . 5 n ) Can the integral test be used to test convergence? A. no, because the terms in the series do not decrease in magnitude B. no, because the terms in the series are not all positive for n c , for some c > 0 C. no, because the series is not a geometric series D. no, because the terms in the series are not recursively defined E. no, because the terms in the series are not defined for all n F. yes B. n = 1 cos ( n ) n 2 Can the integral test be used to test convergence? A. no, because the terms in the series do not decrease in magnitude B. no, because the terms in the series are not all positive for n c , for some c > 0 C. no, because the series is not a geometric series D. no, because the terms in the series are not recursively defined E. no, because the terms in the series are not defined for all n F. yes Solution: SOLUTION Recall that the integral test can be used when we consider a n and the function a n = f ( n ) is decreasing and positive for n c . (Where c is some number that is fixed for the given f ( n ) ). Thus, for n = 1 ln ( 0 . 5 n ) , the integral test cannot be used, be- cause the terms in the series do not decrease in magnitude, and for n = 1 cos ( n ) n 2 , the integral test cannot be used, because the terms in the series are not all positive for n c , for some c > 0. Answer(s) submitted: (incorrect) Correct Answers: A B 5. (1 point) Compute the value of the following improper integral. If it converges, enter its value. Enter infinity if it di- verges to , and -infinity if it diverges to - . Otherwise, enter diverges.
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