In each of the following, supposer< -1. i. Prove that {r"} does not converge to any real number. IHINT: Suppose the sequence does converge to a real number and arrive at a contradiction. The following may be useful. • If {an} converges to a, then {|a»]} converges to {la|}. (Proved in class.) • Parts (a) and (c) of this problem. ii. Prove that lim r" # ∞. n00 iii. Modify the definition above for lim an o to give a definition for the %3D statement lim an o and prove that lim r" + -00.
In each of the following, supposer< -1. i. Prove that {r"} does not converge to any real number. IHINT: Suppose the sequence does converge to a real number and arrive at a contradiction. The following may be useful. • If {an} converges to a, then {|a»]} converges to {la|}. (Proved in class.) • Parts (a) and (c) of this problem. ii. Prove that lim r" # ∞. n00 iii. Modify the definition above for lim an o to give a definition for the %3D statement lim an o and prove that lim r" + -00.
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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![Definition: Let {an} be a sequence of real numbers. We say that lim a, = 00 if,
for every M > 0, there is an NEN such that, if n > N, then an > M.
In each of the following, supposer< -1.
i. Prove that {r"} does not converge to any real number. IIINT: Suppose the
sequence does converge to a real number and arrive at a contradiction. The
following may be useful.
• If {an} converges to a, then {la,]} converges to {lal}. (Proved in class.)
• Parts (a) and (c) of this problem.
ii. Prove that lim r" + 0.
iii. Modify the definition above for lim a, = o to give a definition for the
0o and prove that lim r" -0o.
statement lim an](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87a5eb21-df18-43d0-b53f-a372e6adca02%2Fac3f7a8b-f3fc-476c-9ab7-7990da61fc43%2Fl72v6ko_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Definition: Let {an} be a sequence of real numbers. We say that lim a, = 00 if,
for every M > 0, there is an NEN such that, if n > N, then an > M.
In each of the following, supposer< -1.
i. Prove that {r"} does not converge to any real number. IIINT: Suppose the
sequence does converge to a real number and arrive at a contradiction. The
following may be useful.
• If {an} converges to a, then {la,]} converges to {lal}. (Proved in class.)
• Parts (a) and (c) of this problem.
ii. Prove that lim r" + 0.
iii. Modify the definition above for lim a, = o to give a definition for the
0o and prove that lim r" -0o.
statement lim an
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