The Divergence Test for infinite series (also called the "n-th term test fordivergence of a series") says that:lim12 → ∞←0# UngConsider the seriesNotice that this test tells us nothing aboutn=1situation the series might converge or it might diverge.12 1Σ an divergesn=1121012The Divergence Test tells us this series:an if lim an 0; in that12 →∞0might converge or might divergedivergesconverges= Consider the seriesn=14n³4n³ + 4Based on the Divergence Test, does this series Diverge?O DivergesO Inconclusive

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Answered: The Divergence Test for infinite series…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider the series Σ(n+1)7²n+1 In this problem you must attempt to use the Ratio Test to decide whether the series converges. Compute L = lim n-00 an+1 an Enter the numerical value of the limit L if it converges, INF if it diverges to infinity, -INF if it diverges to negative infinity, or DIV if it diverges but not to infinity or negative infinity. L = Motor in order Which of the following statements is true? A. The Ratio Test says that the series converges absolutely. B. The Ratio Test says that the series diverges. C. The Ratio Test says that the series converges conditionally. D. The Ratio Test is inconclusive, but the series converges absolutely by another test or tests. E. The Ratio Test is inconclusive, but the series diverges by another test or tests. F. The Ratio Test is inconclusive, but the series converges conditionally by another test or tests. Enter the letter for your choice here:
The issue of convergence or divergence of some series can be answered by several different approaches (series tests), and each would give the answer independently. k2 Consider the series E- Answer the questions below regarding some of the tests one might try in order to determine whether the series converges or diverges. k = 1 (e) Would the ratio test be conclusive here? Select the correct choice below and, it necessary, till in the answer box within your choice. O A. Applying the ratio test gives r= 1. Therefore, the ratio test is inconclusive. O B. Applying the ratio test gives r= (type an exact value, or o or – o0). Since r 1, the series DIVERGES by the ratio test. (f) Can the alternating series test be applied here? Select the correct choice below and, if necessary, fill in the answer box within your choice. For this one, we let a, be the m O A. This is an alternating series and hence the alternating series test can be applied. It is clear that the terms a, are decreasing and that…
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Given: Ak = (1 + 2/k)* For both of the following answer blanks, decide whether the given sequence or series is convergent or divergent. If convergent, enter the limit (for a sequence) or the sum (for a series). If divergent, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, or DIV otherwise. (a) The series (A*). k=1 (b) The sequence {A}. \
7a
5. Consider the series n! n=0 for real numbers x. (a) Use the ratio test to determine the values x for which the series is absolutely convergent. (b) Does this mean that the series is convergent for these values of x? Explain. (c) Explain how one can use part (b) above to determine the limit of the sequence {an} where 22n an n!
Press to e (– 1)" Consider the series n7/6 n=1 1. Which of the following statements is true? (You may select more than one if appropriate. No partial credit is awarded for partially correct answers.) (- 1)" ! O The terms of the sequence are nonnegative. n7/6 n=1 1 O The terms of the sequence are decreasing. n7/6 n=1 O The Alternating Series Test may be applied to the series. O The given series is an alternating series. (– 1)" O The terms of the sequence are decreasing. n7/6 n=1 O The Integral Test may be applied to the series. O None of the above 2. 1 lim n→ o n7/6 3. Which of the following statements is true? (You may select more than one if appropriate.) OThe series converges conditionally. O The series converges absolutely. O The series diverges. O None of the above.
3) Ž (-e)" x"+1 What values of x make the series convergent, absolute convergence, conditional convergence, or divergence of the following series?

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