(c) (5 marks) Give an example of i. a sequence of real numbers that is strictly...
41. Let (an) be a sequence of strictly positive real numbers and Sn = ak (a) Suppose that the series Σ an/ S,. an is convergent, determine the nature of the series an is divergent, show that 00 (b) Suppose that the series 1 1 Sn-1 Sp an a/S Then deduce the nature of the series 41. Let (an) be a sequence of strictly positive real numbers and Sn = ak (a) Suppose that the series Σ an/ S,. an...
H-4-2. [5 marks] (a) Consider the sequence (2n 1)I and determine whether it is (eventually) (strictly) increasing or (eventually) (strictly) decreasing or not monotonic. YOU DON'T HAVE TO COMPUTE ANY LIMITS (b) Consider the sequence [10n -3"} and determine whether it is (eventually) (strictly) increasing or (eventually) (strictly) decreasing or not monotonic. YOU DONT HAVE TO COMPUTE ANY LIMITS Grading. M2 S1 R1 Cl
Give an example of infinitely many sets of real numbers, called such that all four conditions are satisfied at once. They are: i) each set is bounded above. ii) for all m and all n. iii) the intersection of and is empty whenever m and n are not equal. iiii) for all n, is not an element of . Not sure what to do here, but I believe it can be done using the fact that there is infinitely many...
6. Give an example of a non-constant sequence that satisfies the given conditions or explain why such a sequence does not exist: (1) {an} is bounded above but not convergent. (2) {an} is neither decreasing nor increasing but still converges. (3) {an} is bounded but divergent. (4) {an} is unbounded but convergent. (5) {an} is increasing and converges to 2.
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
Question 1 3+cos(n) 2n X Which of the following properties hold for the sequence an for n 2 1? l. Bounded Il. Monotonic IIl. Convergent Selected Answer a. I only a. I only b. Il only c. I and Il only d. I and Ill only e. I, II, and III Remember what these conditions mean: Bounded means all terms of the sequence have to lie within a specific range of values. Monotonic means the sequence is ALWAYS increasing or...
5 Consider the following continued fraction 2 + (i) Write the above continued fraction as the limit of a sequence. Also write a recurrence relation between the terms of the sequence. (ii) Show that the sequence is bounded. (i) Show that the subsequence of odd-indexed terms and even-indexed terms are monotonic. (iv) Show that the above continued fraction converges and find the limit. 5 Consider the following continued fraction 2 + (i) Write the above continued fraction as the limit...
Let an, hen be a strictly monotonic sequence of real numbers with aro and the limit of 1. Let Yn be a sequence of Continuous real-valued fmetions on (0,13 with Yn 20 and Jual, so that the support of Yn is in Jan, anta [ A function f : [0,13? → R is defined as: nein | 6%9) - 2(--) -.-) . . a) Cubabe si Cinsel)aly and . (5 testube. Tip: Fix xe [an, Amen] (then overy and do...
7.10 please e) divergence at I = -5? Exercise 7.10. Show that if the sequence and is bounded then the power series > .7 n=0 converges absolutely for p<1. Exercise 7.11. Let A be a set of real numbers with the following property: For every real number Il i) if I, E A then I e A for every I such that I< 1:1), and ii) if I & A then I ¢ A for every I such that :|...
Question 2 (12 marks) (a) Consider the sequence with terms 2n3 5"5 log n , n = 1,2,3,.... an 13 n8n (i) Determine whether {an} converges or diverges. If the sequence converges, find its lmit (ii) Determine whether diverges. Justify your answer an COnverges or n-1 (b) Consider the series (2n)! 2" (n!)? n=1 and determine whether it converges or diverges. Justify your answer Question 2 (12 marks) (a) Consider the sequence with terms 2n3 5"5 log n , n...