3. Consider the sequence (x,) with x, =3 defined recursively by the ruleX 4-x Explore the...
show all work | 2n-1) 2. Consider the sequence |(n+1)! a) is the sequence monotone increasing or monotone decreasing or neither? b) Find upper and lower bounds for the sequence. c) Does the sequence converge or diverge? (Explain) 3. Determine if the series converges or diverges. If it converges, find its sum. => [-1-] c) Ë ?j? – 1-1 j? +1
Xo Xo Problem 1. Show that the recursively-defined sequence x*i-x, - gives the sequence of x-values described in this procedure, as follows: (a) Write the linear approximation 1 (x) to the curve at the point (Xn,f(xn). (b) Find where this linear approximation passes through the x-axis by solving L(x)0 for x. xn + 1-1,-I n). is the recursion formula for Newton's Method. : Xo Xo Problem 1. Show that the recursively-defined sequence x*i-x, - gives the sequence of x-values described...
1. Consider the sequence defined recursively by ao = ], Ant1 = V4 an – An, n > 1. (a) Compute ai, a2, and a3. (b) For f(x) = V 4x – x, find all solutions of f(x) = x and list all intervals where: i. f(x) > x ii. f(x) < x iii. f(x) is increasing iv. f(x) is deceasing (c) Using induction, show that an € [0, 1] for all n. (d) Show that an is an increasing...
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...
Problem 5 A sequence {an) is defined by ay = 1 and an+1 = 3 - Use the Principle of Mathematical Induction (PMI) to show that an is increasing and bounded above by 3 and explam the sequence converges. Using the fact that any converges and an+1 = 3 - find the value of the limit limn+an.
Question 4 can have more than 1 answer 4) The Comparison Test a) is a consequence of the Monotone Convergence Theorem. b) applies only to positive series. c) shows that if o San <br and if į bn converges, then an converges. d) None of the above. n=1 n=1 5) The Divergence Test n=1 a) shows that if lim n = 0, then į an converges. b) applies only to positive series. c) can be used to show that Enti...
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...
Consider the sequence {an} defined recursively as: a0 = a1 = a2 = 1, an = an−1+an−2+an−3 for any integer n ≥ 3. (a) Find the values of a3, a4, a5, a6. (b) Use strong induction to prove an ≤ 3n−2 for any integer n ≥ 3. Clearly indicate what is the base step and inductive step, and indicate what is the inductive hypothesis in your proof.
Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
Assume that the sequence defined by a1 = 3 an+1 = 15-2·an is decreasing and an > O for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)