11 points Problem 3. (3+3+5 pts) Consider the sequence: n sinn n3+1 an a. Is the...
a and an+1= 5an +3 for any natural (Total 5+10= 15 pts) 4. For a positive real number a, consider the sequence (an)1 defined by a1 number n. Answer each queestion. (a) Without using e-N argument, show that the sequence (an)1 converges. (5 pts) (b) Using definition of limits, i.e., using e-N argument, show that the sequence (an)1 is a convergent sequence. If it converges, determine also the limit (10 pts)
a and an+1= 5an +3 for any natural (Total...
Question 2 (12 marks) (a) Consider the sequence with terms 2n35"5 log n n 1,2,3,.. 13 n8n (i) Determine whether ah diverges. If the sequence converges, find its converges or limit. o0 (ii) Determine whether r diverges. Justify your ansv swer an Converges o n-1 (b) Consider the series (2n)! 2 (n!) and determine whether it converges or diverges. Justify your answer IM8 8
Question 2 (12 marks) (a) Consider the sequence with terms 2n35"5 log n n 1,2,3,.. 13...
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...
0-11 points RogaCalcET3 10.4.027. 8. Determine convergence or divergence by any method. Σ-7 -n3/3 7n n e n=1 The series converges The series diverges.
0-11 points RogaCalcET3 10.4.027. 8. Determine convergence or divergence by any method. Σ-7 -n3/3 7n n e n=1 The series converges The series diverges.
Problem 3. Consider the series: 1 n [ln (n)]4 n=2 a) (6 pts) Use the integral test to show that the above series is convergent. b) (4 pts) How many terms do we need to add to approximate the sum within Error < 0.0004.
2. Consider the following series: (-1)n n In n (a) Is the series convergent? Justify your answer (b) Is the series absolutely convergent? Justify your answer.
(9 marks) Let { ln(n+11) n+3 }n=1 be a sequence. a. Find the first 5 terms of the sequence in the exact form. b. Determine whether the sequence is strictly monotone, monotone, eventually strictly monotone, eventually monotone or neither. Prove it. c. Determine whether the sequence is convergent, and if so, find its limit.
Question 3 (18 points). Determine whether the given sequence converges or diverges. Find the limit of each sequence (including too if appropriate) or state that the limit does not exist (DNE). Prove all of your claims and reference the theorems on sequences that you are using. Hint: The fifth word of this question is sequence, not series! (a) an = (-1)" sinn
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...
Express the sequence {an}=1 as an equivalent sequence of the form {bon}n3- {n} +60 - 4=1 An equivalent sequence is On-3- (Simplify your answer. Do not factor.)