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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3. If the series Eo ak x* converges to the function f(x) on the interval (-2, 2), provide a formula for ag in terms of the function f. k-D0 1
#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.)
Uk 7. Suppose that the series uk converges, and Ev diverges. Show that Σ(uk + vk) and Σ(ukvk) both diverge.
Consider the series an where n=1 (-5n - 3)" (-3n-5)" In this problem you must attempt to use the Root Test to decide whether the series converges. Compute an = Which of the following statements is true? A. The Root Test says that the series converges absolutely. B. The Root Test says that the series diverges. L = lim √√|an| 11-00 Enter the numerical value of the limit L if it converges, INF if it diverges to infinity, MINF if it diverges to negative infinity, or DIV if it diverges but not to infinity or negative infinity. L= ✓ C. The Root Test says that the series converges conditionally. D. The Root Test is inconclusive, but the series converges absolutely by another test or tests. E. The Root Test is inconclusive, but the series diverges by another test or tests. F. The Root Test is inconclusive, but the series converges conditionally by another test or tests. Enter the letter for your choice here:
1. If the series an is convergent and an > 0 for all n, then we also know that n=1 (a) Must be Convergent (b) Must be Divergent (c) Can be either Convergent or Divergent, depending on an M8 n=1 an n E
5. a. Investigate the uniform convergence of the series F(x) = - on [1,1]. b. What can you say about the continuity of F(x) on [-1,1]. Explain.
∞ 1. Let an be a POSITIVE infinite series (i.e. an > 0 for all n ≥ 1). Let ƒ be a continuous function n=1 with domain R. Is each of these statements true or false? If it is true, prove it. If it is false, prove it by providing a counterexample and justify that is satisfies the required conditions.
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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