4. ) Suppose that (an) is a sequence of nonnegative numbers such that lim an = 0. Prove that there is some subsequence (an) of (an) such that n-00 Lk=1 anx Converges. Hint: Use the fact that an → 0 to show you can find a subsequence satisfying an <2-k for all k.

Q: Ex. Show the seguence whose nth term is In (n*) an %3D converges. 5n

A:

Q: 1. Define when a sequence X = (x,) in R is said to converge to x E R. Then prove that the sequence X…

A: The given sequence is: xn=3n+12n+5

Q: Prove that the sequence {cn} converges to c if and only if the sequence {cn- c} converges to 0.

A: To prove that the relation between the limits of the two given sequences

Q: Prove using the formal definition of a limit that the following sequence converges: +00 S-n3 + 2n –…

A:

Q: Suppose that a, > 0 and b, > 0 for n > N (N an integer). If lim, 00 1,¬- (an/bn) = ∞ and Sa,…

A:

Q: Show that the sequence defined by 1 a₁ = 1 an is increasing and an < 3 for all n. Deduce that {an}…

A:

Q: Consider the recursively defined sequence s1 = 1 and sn+1 = 1 / (3-sn) for n  ≥ 1. i. Prove that sn…

A:

Q: Let {an}n=1 be a convergent sequence such that a1 = v5 and an+1= /5an Compute lim an n→∞

A: Given a1 = 5  and  an+1 = 5an

Q: use the appropriate limit laws and theorems to determine the limit of the sequence orshow that it…

A:

Q: 3. Use the limit theorems (or homework from 4.1) to show that lim 2an converges. Hint: Do not use…

A:

Q: For the sequence {an}o with lim an exists, this means the sequence converges. Select one: O True O…

A: First is true and second is false.

Q: Use the definition of the limit of a sequence to prove that (-1)"n' lim = 0. n2 + 1

A: To prove limn→0(-1)nnn2+1=0

Q: 3. Prove that a sequence {a,} converges to the real number a if and only if lim sup a lim inf an =…

A: First suppose sequence  an is converges to a Then we have to prove that limsupan=liminfan=a Fix an…

Q: Consider the following sequence X, = + +3 for n e N. Let L = lim¬0 Xn. Which of the following…

A:

Q: If the sequence {a}, is convergent, then the sequence {x} defined by a₁+as+as++an-s+an = Xn is also…

A:

Q: 2. a. Prove using the formal definition of a limit that the following sequence converges: -n3 + 2n –…

A: This is a problem of sequence.

Q: B. Prove that limx → c f(x) = o if and only if for every sequence aŋ that is in the domain of f and…

A: Consider a function fx. Suppose that limx→cf(x)=∞. That is for all N&gt;0, there exists a δ&gt;0,…

Q: Suppose that {a, } is a monotone sequence such that a, 10 or increases- in this case we can't say…

A: We need to choose correct options.

Q: use the appropriate limit laws and theorems to determine the limit of the sequence orshow that it…

A:

Q: Suppose that a, 2 0 and b, 2 0 for all n> N where N is an integer. If an lim = 00, what can you…

A:

Q: State if the sequence {xn}1 define below is convergent or divergent: fix -1< a < 1, and let xn =…

A: The convergence of the series can be checked by applying the limits to ∞. The convergence of the…

Q: Suppose that the sequence {an} converges to a, where 0 < a < 1. Prove that the sequence {a}…

A:

Q: Suppose that (a, } is a monotone sequence such that a, S11 for all n. Must the sequence converge? If…

A:

Q: (b) Prove that if a sequence converges, then its limit is unique. That is, prove that if limn→0 Sn =…

A: We have to give the proof of the second part i.e., part (b) of problem 7.4.2. We have to show that…

Q: In(1+2e") d) an n

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

Q: Consider the sequence defined by a, = In(2n³ + 2) – In(5n³+2n² + 4). Find the limit of an as n → ∞.…

A:

Q: 3.2.6 Theorem If X (x) then a < lim(x) b. is a convergent sequence and if a S X5bfor all n e N,

A:

Q: 1) Use the rigorous definition of convergence (in other words, an epsilon argument) to prove that…

A: We have to prove that sequence xn  is convergence using rigorous definition for convergence sequence

Q: 1. Recall the definition of the limit of a sequence of real numbers (xn) = (x1, X2, X3, . . .) "(Tn)…

A:

Q: 3. Show that if x1 = 1 and if In+1 Van + 2 for n E N, then the sequence (xn) is monotone and…

A:

Q: 1. Prove the sequence {a,} = (1 +-)" Converges to the limit e*, what about the sequence {b,} = (1…

A:

Q: Suppose (an) is a sequence of positive real numbers such that an+1 Prove that liman (an) even…

A:

Q: Let {an}=1 be a convergent sequence such that a1 = V5 and an+1 V5 an. Compute lim an.

A:

Q: Consider the recursively defined sequence s1 = 1 and sn+1 = 1 / (3-sn) for n  ≥1. i. Prove that sn…

A: Consider the given sequence:

Q: 10. Determine if the following sequence a, = In(n)/In(2n) converges or diverges as limit no. Find…

A:

Q: Determine whether the sequence as n goes to infinity (13(3^n) + 15)/(6(4^n)) is divergent or…

A:

Q: 1.) Determine whether the sequence converges and if it does then find its limit: (п-1)! an = n tan…

A:

Q: a) If lim an = 0, then Σ an converges. n→1

A: The divergence test states that for a series ∑n=1∞an, if limn→∞an ≠ 0 , then ∑n=1∞an diverges . That…

Q: For the sequence {a„}, with lim a, does not exist, this means the sequence diverges. Select one:…

A: 1. Let {an}∞n=0={(-1)n}∞n=0 is an oscillating sequence and for this:an=-1; n is odd1; n is…

Q: 14. Suppose (an) is a bounded sequence such that all of its converging subsequences converge to the…

A:

Q: Suppose that {a,} is a monotone sequence such that a, 8 for all n. Must the sequence converge? If…

A: suppose that an is a monotone sequence such that an≤8 ∀n it is known that a sequence is said to be…

Q: -Let a = 1 and define an+1 = /4a, – I for n 2 1. Show that the sequence {an} converges and find the…

A:

Q: 4).   Use the Ratio Test to determine whether  ∞ n = 1 an  converges, where  an  is given. State…

A: Given : To determine the Ratio Test.

Q: an Show that if lim bn =0 and =1 n- 00 converges then En=1ªn converges too. n=D1

A: We have to solve given problem:

Q: 5. Prove directly from definitions that if (Sn) is a sequence such that lim inf s, lim sup s, then…

A: Explanation of the answer is as follows

Q: an = 0 and bn Σ-1 bp Show that if lim n converges then En-1an converges too.

A: Given that:            limn→∞anbn=0 and ∑n=1∞bn converges. Prove ∑n=1∞an converges.

Q: Let (X,) be a sequence of real numbers. If for some x e R, (X,) converges to poth 7x + 2 and 2x - 13…

A: which of the following statement is true.

Q: Show that the infinite sequence 5. an = √ (n2 + 1) − √ (n2 − 1) (n ≥ 1) converges by showing that it…

A: Given sequence is  an = √ (n2 + 1) − √ (n2 − 1) (n ≥ 1)

Q: an = In(3n2 + 1) – In (n² + 1)

A: For the solution follow the next step.

Q: 2n – 7 Prove that the sequence with nth u, = (a) is monotonic increasing, (b) is bounded above, (c)…

A: Given a sequence un = 2n-73n+2 a) It is monotonically increasing if  un≤un+1 ∀n∈ℕ We need to show…