1- if (x,) Chauchy sequence show that (x,) bounded.

Q: Give an example of a bounded sequence {a„}_1 having exactly 2 limitpoints.

A: we have to find a sequence such that it has two limit points.

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: (Зп + 1)! Find the limit of the sequence an = If the (Зп — 1)! limit does not exist, type "Diverges"…

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Q: Use any method to prove that if X = (xn) is a Cauchy sequence of real numbers, then () is also a…

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Q: Determine the limit of the sequence or show that the sequence diverges by using the appropriate…

A: Given:   To find: The limit of C n  if it exist or to show the sequence diverges by appropriate…

Q: 77. an = (In 3 In 2) + (ln 4 − ln 3) + (ln 5 − ln 4) + - - + (In (n − 1) - ln (n − 2)) + (lnn − ln(n…

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Q: how that every Cauchy sequence {pn} is bounded.

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Q: If the sequences {xn } and {yn } are Cauchy sequences, without using theorem 4.3.12 (Cauchy…

A: Given: Sequences {xn}, {yn} are Cauchy sequences.To prove: {xnyn} is a Cauchy sequence. Proof: 1.…

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

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Q: is every cauchy sequence is convergent

A: is every cauchy sequence is convergent?

Q: converges and find th Yn -2- Use the Monotone Convergence Theorem to show that the sequence defined…

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Q: Prove that y is a negative Cauchy sequence if and only if -y is said to be a positive Cauchy…

A: We need to prove that y is a negative Cauchy sequence if and only if -y is a positive Cauchy…

Q: If the sequence (xn) convergence to x, then the sequence (Vxn) also .x convergence to

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Q: Find the limit of the sequence if it converges; otherwise indicate divergence. 7n in an

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Q: Prove the sequence converges and find its limit.

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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: Let {an} and {bn} be two Cauchy sequences of real numbers. Then show that the sequence {|an − bn|}…

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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: Show that the sequence (x,) = () convergence.

A: We have to use limit n tends to infinity.

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: Every convergent sequence of real numbers is not bounded ylgn İhi

A: Every convergent sequence of real numbers is not bounded.

Q: Use Theorem Bounded monotonic Sequences to show that the sequence with the given nth term converges…

A: Theorem: A bounded monotonic increasing sequence is convergent. Sequence: an=7+1n

Q: Find the limit of the sequence if it converges; otherwise indicate divergence. an = (-1)"

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Q: The sequence{In©}, is ) Converge (ii) Diverge

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Q: 26. an = 2 +0.86)"

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Q: Find the limit of the sequence if it converges; otherwise indicate divergence. 8+ (-1)h an %3D

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Q: A divergent sequence is a Cauchy sequence. ylgn ihi

A: False

Q: A momotone sequence converges if and only if it is bounded ylgn İhi

A: True

Q: Find the limit of the sequence * a, = ī sin(--), n = 1,2, 3, ... 3

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Q: use Theorem 1 to determine the limit of the sequence or state that the sequence diverges.an = 12

A: To use theorem 1 to determine the limit of the sequence.

Q: Show that a convergent sequence is bounded both up and down.

A: Show that a convergent sequence is bounded both up and down.

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: 9. Let {xn}, be a sequence of real numbers. What does the boundedness of this sequence mean?

A: Given xnn=1∞ is a real sequence.

Q: Q. 4: Prove that equivalent norms preserve Cauchy property of sequence?

A: We need to prove that equivalent norms preserve the Cauchy property of sequence. Let us suppose we…

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

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Q: Find the limit of the sequence: (;), ÷ (; + -), ; (; + +), E (E )..

A: We have 1112, 1223+24, 1334+35+36, 1445+46+47+48, ... We need to find the limit of this sequence.…

Q: Find if sequence converges or diverges

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Q: 7,11 Write out the first five terms of the sequence, determine whether the sequence converges, and…

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Q: Prove that the following sequence is cauchy sequence by definition (xn) = Vn +1– Vn Hint :…

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Q: f) {1,0,–1,1,0, –1,1,0,...}

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Q: 4. Determine if the sequence is bounded and if it is monotonic. Determine if the sequence converges…

A: This question is related to sequence and series.

Q: Find the limit of the sequence if it converges; otherwise indicate divergence. an = (-1)" (- %3D

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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: 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: Determine if the sequence (nsin(pi/n)) is convergent or divergent. Find its limit if it is…

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

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Q: Determine the limit of the sequence or show that the sequence diverges. If it converges, find its…

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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:

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.