Please solve exercise 2.1.18, with explanations... all information for the question is provided • Exercise 2.1.18. Let {n} be a sequence, and let a be a real number. Suppose that for every positive , there isan M such that n - x ≤e when n ≥ M. Show that limn→∞ n = x.Exercise 2.2.5. Supposen - cos(n)In :=nUse the squeeze lemma to show that the sequence {n} converges, and find the limit.RemarksThe statement in Exercise 2.1.18 is almost the same as Definition 2.1.2, except that "< " has been replaced by theapparently weaker property "< €." Your task is to explain why this change actually does not matter.In Exercise 2.2.5, the instruction to "find the limit" is not really a separate task, for the conclusion of the squeeze lemmaidentifies the value of the limit.

Answered: • Exercise 2.1.18. Let {n} be a…

• Exercise 2.1.18. Let {n} be a sequence, and let a be a real number. Suppose that for every positive &, there is an M such that n − x ≤ e when n ≥ M. Show that limn→∞ In = x.

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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Question

Please solve exercise 2.1.18, with explanations... all information for the question is provided

• Exercise 2.1.18. Let {n} be a sequence, and let a be a real number. Suppose that for every positive , there is
an M such that n - x ≤e when n ≥ M. Show that limn→∞ n = x.
Exercise 2.2.5. Suppose
n - cos(n)
In :=
n
Use the squeeze lemma to show that the sequence {n} converges, and find the limit.
Remarks
The statement in Exercise 2.1.18 is almost the same as Definition 2.1.2, except that "< " has been replaced by the
apparently weaker property "< €." Your task is to explain why this change actually does not matter.
In Exercise 2.2.5, the instruction to "find the limit" is not really a separate task, for the conclusion of the squeeze lemma
identifies the value of the limit.

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