Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
3. So far in the course, we have only talked about sequences which converge to some real number. In this problem, we will use the following definition: A sequence (of real numbers) {n} is said to diverge to (positive) infinity if for all KER, there exists some MEN such that for all n ≥ M, xn > K. In this case, we abuse notation and write lim n = +∞ n→∞ (a) Write down a corresponding definition for a sequence {n} which diverges to negative infinity, and use it to show that lim -n³ = -∞ n→∞ (b) Suppose {n} is a sequence satisfying xn> 0 for all n € N, and furthermore 1 lim = 0 n→∞ Xn Show that {n} diverges to positive infinity. (c) Show that a sequence {x} is unbounded above (i.e. the set {xn: :n € N} is unbounded above) if and only if {n} has a subsequence {n} which diverges to positive infinity. (Hint: When trying to find a subsequence {n} which diverges to positive infinity, try proving that every p-tail of {n} is also unbounded. Then, construct the sequence inductively in order to make sure nk+1 > nk.)

3. So far in the course, we have only talked about sequences which converge to some real number. In this problem, we will use the following definition: A sequence (of real numbers) {n} is said to diverge to (positive) infinity if for all KER, there exists some MEN such that for all n ≥ M, xn > K. In this case, we abuse notation and write lim n = +∞ n→∞ (a) Write down a corresponding definition for a sequence {n} which diverges to negative infinity, and use it to show that lim -n³ = -∞ n→∞ (b) Suppose {n} is a sequence satisfying xn> 0 for all n € N, and furthermore 1 lim = 0 n→∞ Xn Show that {n} diverges to positive infinity. (c) Show that a sequence {x} is unbounded above (i.e. the set {xn: :n € N} is unbounded above) if and only if {n} has a subsequence {n} which diverges to positive infinity. (Hint: When trying to find a subsequence {n} which diverges to positive infinity, try proving that every p-tail of {n} is also unbounded. Then, construct the sequence inductively in order to make sure nk+1 > nk.)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
See similar textbooks
icon
Related questions
Question
3. So far in the course, we have only talked about sequences which converge to some real number. In this problem, we will use the following definition: A sequence (of real numbers) {n} is said to diverge to (positive) infinity if for all KER, there exists some MEN such that for all n ≥ M, xn > K. In this case, we abuse notation and write lim n = +∞ n→∞ (a) Write down a corresponding definition for a sequence {n} which diverges to negative infinity, and use it to show that lim -n³ = -∞ n→∞ (b) Suppose {n} is a sequence satisfying xn> 0 for all n € N, and furthermore 1 lim = 0 n→∞ Xn Show that {n} diverges to positive infinity. (c) Show that a sequence {x} is unbounded above (i.e. the set {xn: :n € N} is unbounded above) if and only if {n} has a subsequence {n} which diverges to positive infinity. (Hint: When trying to find a subsequence {n} which diverges to positive infinity, try proving that every p-tail of {n} is also unbounded. Then, construct the sequence inductively in order to make sure nk+1 > nk.)
Transcribed Image Text:3. So far in the course, we have only talked about sequences which converge to some real number. In this problem, we will use the following definition: A sequence (of real numbers) {n} is said to diverge to (positive) infinity if for all KER, there exists some MEN such that for all n ≥ M, xn > K. In this case, we abuse notation and write lim n = +∞ n→∞ (a) Write down a corresponding definition for a sequence {n} which diverges to negative infinity, and use it to show that lim -n³ = -∞ n→∞ (b) Suppose {n} is a sequence satisfying xn> 0 for all n € N, and furthermore 1 lim = 0 n→∞ Xn Show that {n} diverges to positive infinity. (c) Show that a sequence {x} is unbounded above (i.e. the set {xn: :n € N} is unbounded above) if and only if {n} has a subsequence {n} which diverges to positive infinity. (Hint: When trying to find a subsequence {n} which diverges to positive infinity, try proving that every p-tail of {n} is also unbounded. Then, construct the sequence inductively in order to make sure nk+1 > nk.)
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
SEE MORE TEXTBOOKS