let (sn) be a convergent sequence, and suppose lim sn > a. Prove there exists a number N such that n > N implies sn >a.

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

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Q: Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit.…

A: Answer is infinity

Q: Suppose that (Sn) and (t„) are convergent sequences with lim s, = s and lim t, = t. Then b) lim…

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

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Q: Write out the first five terms of the sequence (-1)"+1 (n+5) determine whether the sequence…

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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}…

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Q: Show directly from definition that if a, is a positive real sequence with lim a, = 0 and lim f(r) =…

A: Solve.

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

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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: 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: because 01. > 1. 1. en n-00

A: If x&gt;1 then limn→∞xn does not exists.

Q: Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit.…

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Q: Find the limit if the sequence converges and lim sup and lim inf if the sequences diverges (c and d)

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Q: E. Suppose that the S, satisfies both lim,-o0 S20 = -1 and lim,-00 S2n+1 = -1. (That is, the…

A: A sequence an has the limit L if for every ∈&gt;0 there is a corresponding integer N such that…

Q: Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit.…

A: We have given a sequence ,  xn = 31 + 19 arctann17n If we say that sequence xn is convergent , it…

Q: Suppose that is a convergent sequence and {an} 3n a, = 12· lim 2n+1) Find lim an. O48 OB. 18 OC12…

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Q: Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit.…

A: For given sequence we have to check its convergence first we have to check    limn -&gt; ♾ | (-1)n…

Q: Given that lim n2(x-5)" converges by the Ratio test. If xEZ then the numerical 1- 00 value of x is…

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Q: Assume that the sequence converges and find its limit. 72 a 1 = 5, a n+ 1- I+ an

A: Given that the sequence converges and the sequence is defined as a1=5, an+1=721+an. Let L be the…

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

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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,)…

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Q: Given that lim n2(x- 5)" converges by the Ratio test. If x EZ then the numerical n- 00 value of x is…

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

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Q: Suppose the sequence {a,} is convergent, in particular lim,-o an = L. Does this mean that every…

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Q: Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit.…

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Q: Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit.…

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Q: Suppose xn2 0 for all n E N and the sequence (-1)"xn converges. What is its limit?

A: A sequence represented by say bn; if converges thenlimn→∞bn=0. It is important to note that the…

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: Suppose that {a,} is a monotone sequence such that an 14 or increases - in this case we can't say…

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Q: Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit.…

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Q: By examining the limit as n→0, the sequence {2"}D=1 O Diverges O Converges to 0 1 Converges to

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Q: Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit.…

A: Consider the sequence an with an=7+2arctann6n. We evaluate the limit limn→∞an as follows.…

Q: Determine if the following sequence converges or diverges and find the limit if possible. Sin(n+ 1)…

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Q: Again, consider the sequence from question 1, S2 4 6 8 10 12'4'8' 16' 32 Use your answer from…

A: The given sequence,       22,44,68,816,1032,........

Q: Use the fact that e = lim (1+ )" and the formula n00 3. 1-() () + – n n+1 to compute the limit of…

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Q: 5. Show that the infinite sequence a, = Vn2 + 1 – yn² – 1 (n 1) converges by showing that it is…

A: Given sequence is  an= √(n^2+1) - √(n^2-1)

Q: Suppose 1 aj = 2 3 4 5 aɔ = az = as = 1 1 3 4 4 5 6. 6. a) Find an explicit formula for a,: b)…

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

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Q: A. Consider the recursively defined sequence s1 = 1 and s+1 = 1/ (7 - Sn) for n > 1. i. Prove that…

A: To show sequence sn is convergent, we used the monotone convergence theorem, which says that every…

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: Write out the first five terms of the sequence with, In(n) determine whether the sequence n+1 n=1…

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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: +oo 19 In n n=1 Determine whether the sequence converges, and if so find its limit. NOTE: Enter…

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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…

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.