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.)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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

Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,