} be a sequence of real numbers. } is unbounded, then {xn} has no limit. 2} is not monotone, then {n} has no limit. 2} converges, then there exists N E N such that |rN+1 – N < 1/2N. xn < 1/2" for all n E N, then {xn} converges. = 1 and xn+1 = xn + 1/n for n > 1, then {x,} is bounded.

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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Determine whether the statement is true or false. If true, provide a proof; if false,
provide a counterexample.
Let {xn} be a sequence of real numbers.
(a) If {xn} is unbounded, then {xn} has no limit.
(b) If {xn} is not monotone, then {xn} has no limit.
(c) If {xm} converges, then there exists N E N such that rN+1 – xN| < 1/2N.
(d) If |æn+1
- xn < 1/2" for all n E N, then {xn} converges.
(е) If 21
= 1 and xn+1 = Xn + 1/n for n > 1, then {xn} is bounded.
Transcribed Image Text:Determine whether the statement is true or false. If true, provide a proof; if false, provide a counterexample. Let {xn} be a sequence of real numbers. (a) If {xn} is unbounded, then {xn} has no limit. (b) If {xn} is not monotone, then {xn} has no limit. (c) If {xm} converges, then there exists N E N such that rN+1 – xN| < 1/2N. (d) If |æn+1 - xn < 1/2" for all n E N, then {xn} converges. (е) If 21 = 1 and xn+1 = Xn + 1/n for n > 1, then {xn} is bounded.
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