ii. For each value of a  ∈ ℝ, give an example of a convergent sequence sn with sn > a for all n, but where limn→∞ sn = a.

Q: Let (a,)nen be a bounded sequence and let b= sup{ a, : nƐN } Show that if b is NOT a member of { a,…

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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: 1. For each of the following sequences, determine if it converges or diverges. If it converges,…

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Q: (a) Use the Bernoulli inequality to show that for all n ∈ N: 1 ≤ n1/n ≤ (1+√n)2/n ≤ (1 + 1/√n)2 b)…

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Q: Prove that if {a,} is a convergent sequence, then to every posi- tive number ɛ there corresponds an…

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Q: Prove that the sequence (xn) = (-1)"+1 [3 .does not converge to any real number

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Q: 2. Determine whether the following sequence converges, and if so, find its limit. (a) nn (b) {n°e-»}…

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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: Let [[x]] be the greatest integer less than or equal to x. Forexample, [[π]] = 3 and [[3]] = 3. For…

A: We have given; [[x]] be the greatest integer less than or equal to x.

Q: Suppose a Cauchy Sequence {xn} is such that for every M ∈ N, there exists a k ≥ M and n ≥ M such…

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Q: Given a sequence (xn) and a real number L , prove or give a counterexample for the following…

A: It is given that:

Q: a Let (n) be a constant sequence in R (i.e. there is some constant c such that In = c for all n).…

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Q: (a) Prove that if the sequence {a„} converges to a, then the sequence {|a„]} converges to |aļ.…

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

A: option  (d)  is correct

Q: 2. Determine whether the following sequence converges, and if so, find its limit. (a) {Vn² + 3n -…

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Q: given parameters a,b > o and the given sequence x₁= b X = n+1 4 (x₂ + 4a) n Xn a. Show that a b.…

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Q: i. For a convergent real sequence sn and a real number a, show that if sn ≥ a for all but finitely…

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Q: 3. For each of the following sequences, prove, using an ɛ,no argument that the sequence converges to…

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Q: i. For a convergent real sequence s, and a real number a, show that if s, 2 a for all but finitely…

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Q: Find the general term of the sequence, starting with n = 1, determine whether the sequence…

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Q: a. In a metric Space (X, d) if (xn) and (yn) are two sequences in X such that : ХX, — хеХапd yn —…

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Q: If a, converges and a, > 0 for every n, show that 4converges. What can you say about a for any…

A: Given: ∑n=1∞an converges and an>0 To prove: ∑n=1∞an2 converges. Proof: ∑n=1∞an2=∑n=1∞an2 Since…

Q: 2. Determine whether the sequence converges or diverges. If it converges, find the limit. (In (In…

A: We need to determine convergence and limit.

Q: Consider the real sequence {an} given by an = In(n + 1) – In(n+ 2), ne Z>0 = NU {0}. Justify whether…

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Q: . Let (xn)neN be a sequence in R. For each n in N, let X1 +x2 + +Xn Yn Show that if (xm)nɛN…

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Q: A) If (xn) is a decreasing and bounded sequence. Prove that (xn) is convergent with lim xn = inf {xn…

A: Multipart: (A)  We need to prove that , if xn is decreasing bounded sequence then , xn is convergent…

Q: 1. Let (n) be an increasing sequence of positive real numbers and define the sequence (yn) as…

A: Given that xn be an increasing sequence of positive real numbers and define the sequence yn as…

Q: Suppose that (an) and (bn) are convergent sequences of real numbers with limits an → a and br → b,…

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Q: For each of the following, prove the sequence converges, and determine the limit. (a) {xn}, where x„…

A: Given that,  zn=0.722.....2 , where,  2 repeats n times. Then,…

Q: For each sequence a, find a number k such that n'an has a finite non-zero limit. (This is of use,…

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Q: We say that a sequence of real numbers (xn)nɛN converges to the real number а, if Ve > 0,3N E N, Vn…

A: Given sequence is xn=-1n+2-n. Definition: A sequence of real numbers xnn∈N converges to the real…

Q: For each sequence an find a number k such that lim n"an is a finite non-zero number. Then use the…

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Q: Suppose that the sequence {an} converges to a   and that a>0. Show that there is an index N such…

A: Prove the following as follows.

Q: 20

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Q: Suppose that the sequence {an} converges to a and that la| < 1. Prove that the sequence {(an)"}…

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Q: Let a < b be real numbers and assume a sequence (Tk)ž is strictly decreasing and Xk E (a, b) for…

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Q: 4. If {Cn} is a convergent sequence of real numbers, does there necessarily exist R > 0 such that…

A: Given that cn is a convergent sequence of real numbers and let it converge to a limit l. Then by…

Q: Let (x,) be a convergent sequence of real numbers and let (a,) and (b,) be subsequences of (x,) with…

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Q: Find the general term of the sequence, starting withn = 1, determine whether the sequence converges,…

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Q: Find the general term of the sequence, starting with n = 1, determine whether the sequence…

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Q: 1. Consider the sequence {an}, where a, e [a, b] for all n e N. Show that is convergent.

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Q: 2. a) Show that for any positive integer k, the sequence defined by pn = converges linearly to p =…

A: Since you have asked multiple questions, we will solve first question for you. If you want other…

Q: State whether the sequence an = Vn +7 converges and, if it does, find the limit. a) O converges to…

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Q: Find the general term of the sequence, starting with n = 1, determine whether the sequence…

A: The given sequence,  522-12, 632-22, 742-32,.........

Q: For each sequence a, find a number k such that lim na, is a finite non-zero number. Then use the…

A: First I have written the limit comparison test and just use them and find that  Fir problem (1) k=5…

Q: 2. Determine whether the following sequence converges, and if so, find its limit. (a) {v군+ 3m-n} n²…

A: The solution of given problem is followed by

Q: Suppose that the sequence (b converges to 2. Which of the following sequences converges to 4…

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Q: Let x_1 = 1/2 and, for n ≥ 1, x_(n+1) = √xn. Prove that the sequence (xn)^∞_n=1 converges and find…

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Q: Let {B,} C R be a sequence and x E [0,1]. Prove that if |B,,41 – B„| 1 'n+: then {B,} is Cauchy and…

A: We give the detailed solution below correcting a slight typo in the problem.

Q: Prove if ( a n ) is a Cauchy Sequence and ( a n ) has a subsequence that converges to L, prove that…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.