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

Q: Prove that if the sequence (xn)n is convergent, then its limit is unique.

A:

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

A: The sequence an is convergenta1=5 and an+1=5 an

Q: Let x1 = 1 and xn+1 1 for n e N. Show that this sequence is convergent and find its In + 2 limit.

A: I have shown the sequence is bounded above and monotone increasing which implies sequence is…

Q: Let {xn} be a bounded sequence on R. Show that lim = 0. n→+∞ n

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

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

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: 8. Prove (from the definition of a limit of a sequence) that if the sequence {a} tends to limit L as…

A: Hi! Thank you for your question.As per guidelines we can solve only one question at a time in case…

Q: 3. If a sequence (rn) in a metric space X is convergent and has limit r, show that every subsequence…

A: We have a sequence ( xn ) in a metric space X which is convergent with a limit x. Then, we have to…

Q: Let {xn} be a bounded sequence on R. Show that Xn lim = 0. - n→+o n

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Q: Let {xn} be a bounded sequence of real numbers. Prove that {xn} converges if and only if lim inf xn…

A: Proof

Q: Find the limit of the sequence an = In(9n +7) – In n. If the limit does not exist, type "Diverges"…

A: Solution:

Q: Let {xn} be a bounded sequence on R. Show that Xn lim = 0. - n→+∞ n

A: follow next step

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: Prove the sequence converges and find its limit.

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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: Let {an} be a monotonic sequence such that an ≤ 1. Discuss the convergence of {an}. When {an}…

A:

Q: 3] Let (xn) be bounded (not necessarily convergent) sequence, and assume that lim(yn) = 0. Show that…

A: A sequence is said to be bounded if it is bounded above and below by a number. That is, the sequence…

Q: an Show that if lim =0 and E-i b, converges then En=1 converges too. an =1 n+ 00

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

Q: Prove that the following sequence is convergent using the limit definition

A: Solve the following

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: Is {3n/(3n-1)} n=1 to infinity, a convergent sequence in R with the usual metric dR(x,y) = abs(x-y)?

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

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Q: ] Let {an} be a convergent sequence of real numbers and let lim a, = a. n00 (a) Prove that {a,} is…

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: Prove, using the definition of the limit of a sequence, that

A: we know that if, r&lt;1 then limn→∞rn=0 because if r is between negative one to one then rn tends to…

Q: Suppose that {a, } and {b,} are two positive sequences, and define up . u /2n² + bn %3D If lim (a,)…

A:

Q: Prove ca) lim (an t anay t --- + that if Ean Converges, then + antr) = 0, (b) Give o7Convergence…

A: We have to use properties of convergent series first.

Q: Show from first principles (in particular not using any limit laws) that if a sequence (an)ı…

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Q: converges for all * > 0 because lim a - In n 0, for all * > 0- n=2 True False

A: Need to check given statement is true or false.

Q: 2.21, Suppose that {an}n€N is a Cauchy sequence in a metric space X, d suppose there exists a…

A: We need to show xn converges to x

Q: Let ₁1 and In+1 = limit. 1 for n N. Show that this sequence is convergent and find its In + 2

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

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Q: 3. Use the definition of limits of sequence to give a formal proof of lim (Vn+1- yn) = 0. n00

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

A:

Q: If {xn} is a bounded sequence on R. Prove: lim n→+∞

A: Given  sequence xn is bounded  we want  to show that  limn→∞xnn=0

Q: (b) Suppose (L.) is a sequence in R". Prove that lim L, = L exists iff every subsequence (z,), also…

A: see below the answer

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

A:

Q: Let {an} be a sequence of real numbers. Prove: The sequence {an} diverges to -oo if and only if lim…

A: Solution: let us consider the sequence {an} of real numbers. We say {an} diverges to -∞ if and only…

Q: 28. yn = ne/n

A:

Q: If possible, give an example of 2 convergent sequences an and bn such that E(an + bn) diverges.

A:

Q: Use the definition of the limit of a sequence to prove that 3n +1 lim 2n + 5 3 %3D 21

A: We need to use the definition of the limit of a sequence. we have the definition of limit as,…

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: is converges, then lim a, Lan =. If n=1

A: Given that ∑n=1∞an converges then the limn→∞an=∞ If the series converges then the sequence of the…

Q: Let {xn} be a bounded sequence on R. Show that Xn lim = 0. n→+∞ N

A: Given sequence {xn} is bounded we want to show that  limn→+∞xnn=0 we  use the limit limn→∞1n=0

Q: 4. Suppose that lim r, = 0. If (yn) is a bounded sequence, prove that lim In Yn = 0.

A: Bounded sequence: If there is a positive number M such that |Xn|≤M for every n∈N, the sequence xn is…

Q: Use Theorem 4 from Section 10.1 to determine the limit of the sequence, or state that the sequence…

A: The given sequence: an=e6n(10n+13)

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.