2. (30 points) 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 M E N such that for all n ≥ M, în > K. In this case, we abuse notation and write lim n +∞ n→∞ Write down a corresponding definition for a sequence {n} which diverges to negative infinity, and use it to show that lim -n³ = =10 n→∞
2. (30 points) 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 M E N such that for all n ≥ M, în > K. In this case, we abuse notation and write lim n +∞ n→∞ Write down a corresponding definition for a sequence {n} which diverges to negative infinity, and use it to show that lim -n³ = =10 n→∞
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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Transcribed Image Text:2. (30 points) 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 M E N such that for all n ≥ M, în > K. In this case, we
abuse notation and write
lim xn = +∞
n→∞
(a) Write down a corresponding definition for a sequence {n} which diverges to
negative infinity, and use it to show that lim -n³
3
= -∞
n→∞
(b) Suppose {n} is a sequence satisfying än > 0 for all n € N, and furthermore
1
lim
n→∞ Xn
Show that {n} diverges to positive infinity.
=
0
(c) Show that a sequence {n} is unbounded above (i.e. the set {xn ne N} is
unbounded above) if and only if {ïn} has a subsequence {n} which diverges to
positive infinity.
(Hint: The (
) direction is easier, so you may want to start with that.
For the ( ⇒) direction, I'd recommend splitting the proof into parts:
(i) prove that if {ïn} is unbounded, then every p-tail of {x} is also unbounded.
(ii) inductively construct a subsequence {n} which satisfies n >k for all
KEN.
(iii) prove that this subsequence does what you want it to do.)
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