Exercise 8. Let {an}-1 be a sequence such that an lim L+0. n-00 n Prove that {an}1 is not bounded.

Q: 4*++3" Consider the sequence a, = (a) Is the sequence monotonic pr non-monotonic? If it is…

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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: (a) use Theorem 9.5 to show that the sequence with the given nth term converges and (b) use a…

A: Solution:Given      an=7+1n

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

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Q: 1. Give an example of a sequence {an} where lim(an+2 – an) = 0 but {an} is divergent.

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Q: 21

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Q: 10. Use the definition of the limit of a sequence to prove that (3n+1 lim 2n + 5/ for all positive…

A: we have to use definition of limit of sequence.

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

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

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Q: 22: Establish the divergence of the sequence x, =(-1)"

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Q: n2 – 1 - at the sequence ɑn is eventually stric n2 + 2 so show that {a,} is bounded above. What can…

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Q: Assume that the recursively defined sequence converges and find its limit. 10a a, = 10, an+1 (Type…

A: Use limit concept

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

A: follow next step

Q: determine the limit of the sequence or show that the sequence diverges.

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Q: 3. Suppose that the sequence (a,} is monotone, and that the sequence {a)} converges. Prove that the…

A: We have given sequence {an} that is monotonic  and sequence {an^2}  which is convergent. To prove…

Q: The greatest lower bound for the sequence {0.1,0.01,0.001, 0.0001, ...} is

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Q: Assume that the recursively defined sequence converges and find its limit. a1 = - 2, an +1 = /14 +…

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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: a) Show that the sequence a, = (-1)" +- does not converge. b) Let b, be a sequence such that for all…

A: Given the sequence an = ( -1 )n + 1/n . It has two subsequences a2n = 1 + 1/n converging to 1 and…

Q: Assume that the recursively defined sequence converges and find its limit. 72 a, = 1, an +1 1+ an…

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Q: (п +5)(п — 3) 1 | converges. If so, find its limit. n-1 Determine whether the sequence 4n?

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Q: 2. Give an example of an unbounded sequence {an} of negative terms such that an does not diverge to…

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Q: C) Prove that a Cauchy sequence of real numbers is bounded.

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Q: Give an example of a nonincreasing sequence with a limit.

A: The nonincreasing sequence is a sequence {An} of the real numbers that never increases that is An+1…

Q: 2) Show that the sequence defined recursively by 3 4 An aj = 5 аn+1 satisfies 2 < an < 5 and is…

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Q: 1 lim n→∞ n² 0 =

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Q: 1. Let the sequence defined by {(5" +7")*}, be given. Show that the sequence is (a) bounded and…

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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,

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Q: Prove that the sequence (an) = (-1)"+' does not converge to any real number. %3D

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Q: Assume that the recursively defined sequence converges and find its limit. 8 a, = 3, an+1 2+ an The…

A: Solve the following

Q: Show directly from the definition that if (x„) and (yn) are Cauchy sequences, then (X, + yn) and…

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Q: 1. Use the definition of a convergent sequence to show {7 - converges to 7 where {7- is considered…

A: Convergent Sequence in Metric Space: Let be a metric space and let be a sequence in We say that…

Q: Find a sequence of real numbers {a;} such that lim a, does not n-00 xist, but it has 3 subsequential…

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Q: 4. Suppose that lim s, =0 and suppose that (t, ) is a bounded sequence. Show that lim s,t, = 0 %3D

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

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Q: prove that if X =CXn) is bounded increasing sequence of real ConVergent and Lim (xu) - Sup [An ineN]…

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Q: 3] Prove that the sequence (x,) = (-1)"+' does not converge to any real number.

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Q: 28. yn = ne/n

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

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Q: C. Show that any sequence of positive real numbers either has a subsequence that converges, or else…

A: Proof of theorem

Q: Determine whether the sequence converges or diverges an = In(n + 1) – In(n) %3D lim an %3D n- 00

A: Given sequence is an=lnn+1−lnn We have to find the limn→∞an Use property lnm−lnn=lnmn…

Q: a) an en 3n

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Q: Exercise 9. Let {an}_1 be a bounded sequence and {b„}=1 be a sequence converges to 0. Prove that the…

A: Given that an is bounded. Then there exists M&gt;0, such that an≤M for all n. Thus we get  anbn≤Mbn…

Q: 4. Define a sequence by an+1 = -an?, with -1 < ao < 0. Prove that {an} is increasing and bounded…

A: Consider the terms of the sequence put n=0 a1=-a02 here -1≤a0&lt;0 So -1≤a1&lt;0 but certainly a1≥a0…

Q: What theorem sometimes enables us to use l’Hôpital’s Rule to calculate the limit of a sequence? Give…

A: The theorem which enables us to use L'Hospital’s Rule  is,

Q: 1. Prove that a bounded divergent sequence has two subsequences converging to dif- ferent limits.

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

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Q: Use Theorem 1 to determine the limit of the sequence or type DIV if the sequence diverges. 7n An…

A: To find:

Q: 8) Show that the sequence 12, V12 + vi2, v12 Is bounded and monotonic. Then find its limit. 12 V12…

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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.