1. Suppose {an} and {bn} are positive sequences and that there exists N so that n >N = an < bn. Show that if£ bn converges, then so does an.

Q: 18. Which of these sequences is/are not bounded? I. a, = (-1)", n> 1 А. II only В. I only С. П only.…

A: Option (C) is correct. II only.

Q: Exercise 26. Suppose {an} is a sequence which converges to 0, {n} is another sequence, and s, k, and…

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Q: 1. Prove that for any n > 2, n E N, one has (1-)(--)(--) - (-) - n +1 2n

A: I have solved question no. 1 according to policy rule.   To solve q.1, I have used method of…

Q: Let a, = (-1)" + -, vn 2 1. Which of the statements below is/are true? I: Ja,|S 2, Vn 2 1 II: lim,-…

A: There are two properties for a sequence to be convergent, and they are as follows: Monotonicity…

Q: 5. Consider the sequence {an}. Defined by the recurrence relation; a₁ = 1, an+1 = (2an+3), for n ≥…

A: We have to prove the sequence is increasing bounded above by

Q: Suppose that the sequence x0, x₁, x2... is defined by x = 6, x₁ = 3, and xk+2 = 2xk+1+3xk for k≥0.…

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Q: If (an) is a sequence converging to L, with an 20 for all n, prove that L≥ 0 and that (√an)1…

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Q: Suppose {n}1 and {n} are two sequences of real numbers such that {n}1 n=1 converges and {n Yn}=1…

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Q: Show the following statement is true or not with proof. If {n} converges then {[sn]} converges.

A: We have to show the following statement is true or not with proof. If {sn} converges then {|sn|}…

Q: 58. Which of the following sequences is/are not convergent? I: a, := n+1 3n2-4 ,n 2 1 II: a, := n+1…

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

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Q: 43) Suppose that an > 0 for all n and that n=1 anbn necessarily converge? ∞ Σa an converges. Suppose…

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Q: how whether the following infinite sequences converge or diverge : 00 {an} = 2) {bn} = {Vn6 + 12n³ –…

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Q: Suppose that an is a sequence of positive numbers such that Σan converges. Let bn be an arbitrary…

A: Regardless of the sequence bn, the series ∑n=1∞​an​bn​converges. This conclusion holds because we…

Q: 5. Given two Cauchy sequences {an}_1, {bn}_1 in Q. Show that (a) {an + bn}_1 is Cauchy; (b) (anbn)…

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Q: A sequence an converges in the mean if an n 1a converges. Prove that if an → L, then ān → L, but the…

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Q: Consider the sequence {an}, defined by ai = 2 and an =(2an-1+5) for n > 2. %3D Use induction to…

A: Proof by induction for a given statement deals with first proving the statement is true for n=1 and…

Q: A monotone sequence converges if and only if it is bounded.

A: (A) Suppose xn s a monotone convergent sequence.  Since every convergent sequence is bounded,…

Q: Suppose Sanx",Σ bnan converges for all x in (-1,1). Σana" = Σbnan in a subset E of (-1,1) implies…

A: It is given that,  ∑anxn and ∑bnxn are convergent for all x in (-1, 1). Then,  ∑anxn = ∑bnxn in a…

Q: n is bounded and monotonic for all n. n +1 6. Show that the sequence An

A: The objective is to solve the above question.

Q: 6. The sequence (xn) = (3,3,3,3, ...) is .. a) not Cauchy b) unbounded c) divergent d) convergent

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Q: 3. Prove that if (a, } and (b,} are Cauchy sequences of rational numbers satisfying lim (a, - b, = 0…

A: 3.

Q: Let {an}1 and {bn}n=1 be sequences. Prove or disprove: If {an – bn}=1 converges to 0 and {an}-1…

A: Note that, if limn→∞an and limn→∞bn exist, then limn→∞an-bn=limn→∞an-limn→∞bn. It is given that,…

Q: 2) The purpose of this problem is to give you some practice with the most common application of The…

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Q: 5. If {a} is a convergent sequence in R and a = lim a,, show that a = lim inf an lim sup an -

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Q: Does 1/201+1/208+1/227+1/264+... converge or diverge and why?

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Q: Aşağıdaki dizilerden hangisi “ yakınsak, azalan ve en küçük üst sının 1" olan bir dizidir? b) {an}=…

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Q: 2. Let (an)1 be a sequence of integers. Prove that (an) converges if and only if there exists a…

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Q: Give an example of two sequences (an) and (bn) such that all of the following are simultaneously…

A: To Give: An example of two sequences an and bn such that all of the following are simultaneously…

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: Suppose that {an}_₁ is a sequence defined by an = the Squeezing Theorem. We wish to find sequences…

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Q: uppose that the sequence x0, x1, x2... is defined by xo = 3, x₁ =0, x₂=5, and xk+3 = 13xk+1+12x for…

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Q: 43. Which of these sequences is/are not bounded? I: a, 1= (-1)", n 21 A. III only B. I only C. II…

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Q: 13. Let 2 p1 <p2 <. be the sequence of primes in increasing order. Prove that Pn ≤ 22"-1 for all n ≥…

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Q: converge to the same limit.

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Q: a1 = 1, an+1 = 3 - An induction method to prove monotor

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Q: an 57" n! nal

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Q: 8. Suppose ao, a1, a2, ... is a sequence such that ao = a₁ = 1 and, for n ≥ 1, an+1 = n(an + an-1).…

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