Q6. Let X = (0, 3) equipped with the usual metric d.Зп - 1,Consider the sequence {r„} = {-(a) Is {rn} a convergent sequence? Why?(b) Is {r„} a Cauchy sequence? Justify your answer.(c) Is the metric space ( (0, 3), d ) complete?Prove your answer.(d) Answer the previous parts for X = (0, 3].

Given: Sequence xn=3n-1n and X=(0, 3) with usual metric d. TASK: To Explain the following: (a) xn is…

Answered: Q6. Let X= (0, 3) equipped with the…

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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INDEX NUMBÆR:. 1. Let an = (-1)" +!, vn2 1. Which of the statements bekow is/are true? I: [a,|52, Vn 21 II: lim, - sup a, = 1 and lim,-, inf a, = -1 III: a, is a convergent sequence. А. I and II only B.I only C.II only D.I, II and III E None of the above. :400
Find a sequence (rn) with lim sup (xn) = 3 and lim inf (xn) -2. Justify your answer.
The sequence (xm) = (1, 3, 1, 3, 1, . .., 1, 3, ...) is: O Converge to 3. O Divergent. Converge to 0 and 3. Converge to 0. O Converge to 1.
G A.Write the first 5 terms of the sequence defined by the ff. functions. 1.F(W) = 20-1 2F (1) = 30 + 21: 3. (5) : 87 -5 # Fla) = n+1 5. F (1) = nu 준이의 6.F(1) = (-1)727. 7. F(x)=1 8: F(n)= n(nti) 22 9. F(n)= 13. E(7) = (-1)-1 n
Determine if the sequence is increasing, decreasing, not monotonic, bounded below, bounded above and/or bounded. 3 — п (b) {(-3)"}-0 (c) {sin (n)}4 (a) {: n=4 1- 3n n=1
2. Consider the sequence an given by 2n +1 n³ (a) Calculate the first four terms of the sequence (as integers or fractions). an = for n ≥ 1. (b) Is this sequence bounded above or below, or not bounded at all? Explain your answer in detail. Provide bounds if possible. (c) Does this sequence converge or diverge? If it converges, find the limit.
Determine if the sequence is monotonic and if it is bounded. 2"5" an = n! nal Select the correct answer below and, if necessary, fill in the answer box(es) to complete your choice. OA. (a) is monotonic because the sequence is nonincreasing. The sequence has a least upper bound when n = but is unbounded because it has no lower bound. (Simplify your answer.) OB. (a) is not monotonic. The sequence is unbounded with no upper or lower bound. OC. (a) is not monotonic. The sequence is bounded by a greatest lower bound of when n. (Simplify your answers.) and a least upper bound OD. (a) is monotonic because the sequence is nondecreasing. The sequence has a greatest lower bound of but is unbounded because it has no upper bound. (Simplify your answer.)

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Q6. Let X= (0, 3) equipped with the usual metric d. Consider the sequence {n} = {- 3n-1, ·} n (a) Is {n} a convergent sequence? Why? (b) Is {n} a Cauchy sequence? Justify your answer. (c) Is the metric space ( (0, 3), d) complete? Prove your answer. (d) Answer the previous parts for X = (0,3].