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13. If x1 := 2 and x+1 = 2+1/x, for n 2 1, show that (x,) is a contractive sequence. What is its limit?

13. If x1 := 2 and x+1 = 2+1/x, for n 2 1, show that (x,) is a contractive sequence. What is its limit?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Question number (13) only. This material is Real analysis.
are sure that x-x6l 3/(7* - 20) = 243/48 020 < 0.0051. Actually the approximation is substantially better than this. In fact, since |x6- xsl < 0.000 0005, it follows from 3.5.10(ii) that |x" - x6 Sx6 - x5 < 0.000 0004. Hence the first five decimal places of x6 are correct. Exercises for Section 3.5 1. Give an example of a bounded sequence that is not a Cauchy sequence. 2. Show directly from the definition that the following are Cauchy sequences. (1 (a) (b) +...+ 2! 3. Show directly from the definition that the following are not Cauchy sequences. (*). (a) (-1)"), (b) (c) (In n) 4. Show directly from the definition that if (xn) and (yn) are Cauchy sequences, then (x,+ yn) and (XnYn) are Cauchy sequences. = 0, but that it is not a Cauchy sequence. 5. If Xn= n, show that (x,) satisfies lim|xn+1- X| 6. Let p be a given natural number. Give an example of a sequence (Xn) that is not a Cauchy sequence, but that satisfies lim|xn+p - Xn = 0. 7. Let (x) be a Cauchy sequence such that x, is an integer for every n E N. Show that (x) is ultimately constant. 8. Show directly that a bounded, monotone increasing sequence is a Cauchy sequence. 9. If 0<r< 1 and x+1 - x„ < " for all n E N, show that (xn) is a Cauchy sequence. 10. If x1 < x2 are arbitrary real numbers and x, = (Xn-2 + Xn-1) for n> 2, show that (x) is convergent. What is its limit? 11. If yi <y2 are arbitrary real numbers and y, := }yn-1 +?yn-2 for n > 2, show that (y) is convergent. What is its limit? !i! 12. If x1 > 0 and xn+1= (2 + xn)-for n 2 1, show that (x„) is a contractive sequence. Find the limit, 13. If x1 := 2 and x+1 := 2 +1/x, for n > 1, show that (x) is a contractive sequence. What is its limit? 14. The polynomial equation x-5x+1 0 has a root r with 0<r< 1. Use an appropriate contractive sequence to calculate r within 104. Section 3.6 Properly Divergent Sequences For certain purnoses it i oom
Transcribed Image Text:are sure that x-x6l 3/(7* - 20) = 243/48 020 < 0.0051. Actually the approximation is substantially better than this. In fact, since |x6- xsl < 0.000 0005, it follows from 3.5.10(ii) that |x" - x6 Sx6 - x5 < 0.000 0004. Hence the first five decimal places of x6 are correct. Exercises for Section 3.5 1. Give an example of a bounded sequence that is not a Cauchy sequence. 2. Show directly from the definition that the following are Cauchy sequences. (1 (a) (b) +...+ 2! 3. Show directly from the definition that the following are not Cauchy sequences. (*). (a) (-1)"), (b) (c) (In n) 4. Show directly from the definition that if (xn) and (yn) are Cauchy sequences, then (x,+ yn) and (XnYn) are Cauchy sequences. = 0, but that it is not a Cauchy sequence. 5. If Xn= n, show that (x,) satisfies lim|xn+1- X| 6. Let p be a given natural number. Give an example of a sequence (Xn) that is not a Cauchy sequence, but that satisfies lim|xn+p - Xn = 0. 7. Let (x) be a Cauchy sequence such that x, is an integer for every n E N. Show that (x) is ultimately constant. 8. Show directly that a bounded, monotone increasing sequence is a Cauchy sequence. 9. If 0<r< 1 and x+1 - x„ < " for all n E N, show that (xn) is a Cauchy sequence. 10. If x1 < x2 are arbitrary real numbers and x, = (Xn-2 + Xn-1) for n> 2, show that (x) is convergent. What is its limit? 11. If yi <y2 are arbitrary real numbers and y, := }yn-1 +?yn-2 for n > 2, show that (y) is convergent. What is its limit? !i! 12. If x1 > 0 and xn+1= (2 + xn)-for n 2 1, show that (x„) is a contractive sequence. Find the limit, 13. If x1 := 2 and x+1 := 2 +1/x, for n > 1, show that (x) is a contractive sequence. What is its limit? 14. The polynomial equation x-5x+1 0 has a root r with 0<r< 1. Use an appropriate contractive sequence to calculate r within 104. Section 3.6 Properly Divergent Sequences For certain purnoses it i oom
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