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6. Suppose that the sequence {an} converges to a and that a > 0. Show that there is an index N such that an> 0 for all indices n > N.

6. Suppose that the sequence {an} converges to a and that a > 0. Show that there is an index N such that an> 0 for all indices n > N.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 72E
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c. If the sequences {an + b} and {a} converge, then the sequence {bn} also converges. a. d. If the sequence {lan]} converges, then the sequence {a} also converges. 2. Using only the Archimedean Property of R, give a direct e-N verification of the following limits: 1 lim n→∞ √√n a. /n = 0 + 3. Using only the Archimedean Property of R, give a direct e-N verification of the convergence of the following sequences: n +3 b. n = and show that an = 1/2. d. Show that {an} does not converge. j(j + 1) 2 b. CONVERGENT SEQUENCES 4. For the sequence {an} defined in Example 2.3: a. What are the terms a10, a20, 930? b. Find the second index n for which an = 1/4 and the fourth index n for which an = 1. c. For j an odd natural number, set + Hint: Define a₁ = 1 = 0 lim n→∞n +5 33 n² n² + j+1 2 5. For the sequence {an} defined in Example 2.3 and any rational number x in the interval (0, 1], show that there are infinitely many indices n such that an = x. 6. Suppose that the sequence {an} converges to a and that a > 0. Show that there is an index N such that an> 0 for all indices n > N. 7. Suppose that the sequence {an} converges to & and that the sequence {b} has the property that there is an index N such that lim n/n = 1. 84x an = bn for all indices n ≥ N. Show that {b} also converges to l. (Suggestion: Use the Comparison Lemma for a quick proof.) 8. Prove that the sequence {cn} converges to c if and only if the sequence {cn- c} converges to 0. 9. Prove that the Archimedean Property of R is equivalent to the fact that limn→∞ 1/n = 0. 10. Prove that nl/n - 1 and use the Binomial Formula to show that for each
Transcribed Image Text:c. If the sequences {an + b} and {a} converge, then the sequence {bn} also converges. a. d. If the sequence {lan]} converges, then the sequence {a} also converges. 2. Using only the Archimedean Property of R, give a direct e-N verification of the following limits: 1 lim n→∞ √√n a. /n = 0 + 3. Using only the Archimedean Property of R, give a direct e-N verification of the convergence of the following sequences: n +3 b. n = and show that an = 1/2. d. Show that {an} does not converge. j(j + 1) 2 b. CONVERGENT SEQUENCES 4. For the sequence {an} defined in Example 2.3: a. What are the terms a10, a20, 930? b. Find the second index n for which an = 1/4 and the fourth index n for which an = 1. c. For j an odd natural number, set + Hint: Define a₁ = 1 = 0 lim n→∞n +5 33 n² n² + j+1 2 5. For the sequence {an} defined in Example 2.3 and any rational number x in the interval (0, 1], show that there are infinitely many indices n such that an = x. 6. Suppose that the sequence {an} converges to a and that a > 0. Show that there is an index N such that an> 0 for all indices n > N. 7. Suppose that the sequence {an} converges to & and that the sequence {b} has the property that there is an index N such that lim n/n = 1. 84x an = bn for all indices n ≥ N. Show that {b} also converges to l. (Suggestion: Use the Comparison Lemma for a quick proof.) 8. Prove that the sequence {cn} converges to c if and only if the sequence {cn- c} converges to 0. 9. Prove that the Archimedean Property of R is equivalent to the fact that limn→∞ 1/n = 0. 10. Prove that nl/n - 1 and use the Binomial Formula to show that for each
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