14.9 The convergence of a series does not depend on any finite number ofthe terms, though of course the value of the limit does. More precisely,consider series an and Σbn and suppose the set {n € N : an ‡bn}is finite. Then the series both converge or else they both diverge.Prove this. Hint: This is almost obvious from Theorem 14.4. 14.4 Theorem.A series converges if and only if it satisfies the Cauchy criterion.

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Answered: 14.9 The convergence of a series does…

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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This question concerns the following exercise. Exercise Determine whether the sequence (an) given by n.³ n.³ an = 3n²2 - 2 3n² - 1 converges, and find its limit if it does. Name any results or rules that you use. You may use the basic null sequences listed in Theorem D7 from Unit D2. (a) Explain why the following solution to this exercise is incorrect and/or incomplete, identifying three errors or significant omissions. For each error or omission, explain the mistake that the writer of the solution has made. (There may be more than three errors or omissions, but you need identify only three. These should not include statements or omissions that follows logically from earlier errors or omissions.) Solution (incorrect and/or incomplete!) For n=1,2,..., we have n.³ 3n2 - 2 n³ 3n² - 2 23 3n² - 2 |an| 73 VI = (since n.³ 3n² 2n 3 + n³ 3 3n²-1 + n³ 3n2 - 1 n³ 3n2 - 1 + n³ 3n² - 2 n³ 3n² > 0 and (by the Triangle Inequality) n.³ 3 3n² - 1 > 0) Now 2n/3→ ∞o as noo, so by the Squeeze Rule…
Let r > 0 be a positive real number. This problem will give a simple approach to limn→∞ rn.i. Explain briefly why an = rn can be written recursively as a0 = 1, an = r ⋅ an-1 for n > 0.ii. Using Problem A and (i), solve for the possible limits of an if the sequence converges.iii. Show that an is monotone. (It may be increasing or decreasing, depending on r.)iv. Combine (ii) and (iii) to show limn→∞ rn converges to 0 for r < 1, converges to 1 for r = 1, and diverges to ∞ for r > 1.
3. Prove each of the following. (a) If Ea converges, then i must diverge. 1+ a. (b) If ar 20 for all k e N and ar converges, then must converge. 1+ ak 1 (c) The series 3k + k – 1 k=1 converges and its sum is at most 2
#76 part a
Y.7.
Using only the e – N definition of convergence prove that the sequence (an)nEN given by the formula 5n7 7n an = converges to 0. (Hint: It may be helpful to prove first an inequality that says that the geometric quantity in the denominator grows a lot faster than the polynomial in the numerator.)
4). Use the Ratio Test to determine whether ∞ n = 1an converges, where an is given. State if the ratio test is inconclusive.∞ n = 134n(n!)4(4n)!Identify an. Evaluate the following limit. lim n → ∞ an+1an Since lim n → ∞ an+1an 1, . please show step by step .
#61
A sequence (n)-1 is said to oscillate if lim inf xn
i. Suppose that the sn satisfies both limn→∞s2n = 3 and limn→∞s2n+1 = 3. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 3.) Show that also limn→∞sn = 3.ii. Give an example of a sequence where the sequences given by the even and by the odd terms both converge, but where the entire sequence does not converge.

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