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Assume 0 ≤ a ≤ b. Does the sequence {(a + b)=1 diverge or converge? If the se- quence converges, find the limit. the sequence

Assume 0 ≤ a ≤ b. Does the sequence {(a + b)=1 diverge or converge? If the se- quence converges, find the limit. the sequence

Advanced Engineering Mathematics
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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Could you do 43 and 45, thanks

abou quence converges, find the limit. 43. Assume 0 ≤ a ≤ b. Does the sequence {(a + b)¹/"}=1 diverge or converge? If the se- AQ 44. Does the sequence k 1 { (√²+)} k² + n k=1 diverge or converge? If the sequence converges, find the limit. 45. Show that if x is any real number, there is a sequence of rational numbers converging to x. *46. Show that if x is any real number, there is a sequence of irrational numbers converging to x. 47. Suppose that {an)=1 converges to A and that B is an accumulation point of {an: nEJ). Prove that A = B. MISCELLANEOUS 48. Suppose that {an)n-1 and (bn)n-1 are two sequences of positive real numbers. We say that an is O(bn) (read as an is "big oh" of b) if there is an integer N and a real number M such that for n ≥ N, an ≤ M. bn. Prove that if {an/bn)n-1 converges to L # 0, then an is O(bn) and bn is O(an). What can you say if L = 0? Illustrate with examples. 194 PROJECT 1.1 The purpose of this project is to prove that the sequence 8 {(₁ +9) T n=1 imply 11 TOY 2.3 To song T
Transcribed Image Text:abou quence converges, find the limit. 43. Assume 0 ≤ a ≤ b. Does the sequence {(a + b)¹/"}=1 diverge or converge? If the se- AQ 44. Does the sequence k 1 { (√²+)} k² + n k=1 diverge or converge? If the sequence converges, find the limit. 45. Show that if x is any real number, there is a sequence of rational numbers converging to x. *46. Show that if x is any real number, there is a sequence of irrational numbers converging to x. 47. Suppose that {an)=1 converges to A and that B is an accumulation point of {an: nEJ). Prove that A = B. MISCELLANEOUS 48. Suppose that {an)n-1 and (bn)n-1 are two sequences of positive real numbers. We say that an is O(bn) (read as an is "big oh" of b) if there is an integer N and a real number M such that for n ≥ N, an ≤ M. bn. Prove that if {an/bn)n-1 converges to L # 0, then an is O(bn) and bn is O(an). What can you say if L = 0? Illustrate with examples. 194 PROJECT 1.1 The purpose of this project is to prove that the sequence 8 {(₁ +9) T n=1 imply 11 TOY 2.3 To song T
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