Problem 2. i) Show that if {n} diverges and xo is any number, then there exists ɛ and a subsequence {n} such that xnk − xo| > ɛ for all k ii) Let {n} be a sequence that converges to a number xo. Show that there exists a subsequence {n} with |¤nk − xo| < 1/10k for all k. In other words, some sequences converges rapidly to xo.

Q: Prove that the sequence {x} defined recursively as follows is convergent xo = 15 =3---|| Xn+1 = 3-

A: A sequence {xn } is given. If we show that the sequence is a Cauchy sequence then the sequence is…

Q: 5. The Fibonacci sequence is defined recursively as an+2 = an+1 + an, with a₁ = 1 = a2. (a) Write…

A: Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: PIUVE thlat Un 2 X9. A sequence {an} is defined recursively by ao = 1, a1 = 2, a2 = 24, and an = n 2…

A: The recursive relation has been proved in step 2.

Q: Show that the sequence defined by  a sub 1= 1  and  a sub n + 1 = 3 - (1/ a sub n) is increasing and…

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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: Suppose that the sequence {n} is such that xn 20 for all n eN and that lim(-1)"xn exists. Show that…

A: We will use Triangle inequality.

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: Prove that if {xn} is a sequence such that 1+ n |æn| < for all n E N, 1+ n2 then the sequence {xn}…

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Q: 4. Let a₁ = √2, and an+1 = √2+ an Vn € N. Show that {a} is a monotone sequence by mathematical…

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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: 4) Consider the recursive sequence defined by: x1 = 3 x₂ = 5 In = 3xn-1 + 2xn-2 for n ≥ 3. Prove…

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Q: Suppose that the sequence x0, x1, x2... is defined by xo = 5, x₁ = 5, and xk+2 = −8xk+1-15xk for…

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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: Find a function  f(n)  that identifies the nth term  an  of the recursively defined sequence, as  an…

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Q: Let an be a sequence defined by a1 = 0 and a2 = 1 and for n > 3 an = (n – 1)(an-1+an-2). %3D

A: Here, the terms given in the series are, a1=0,a2=1an=n-1an-1+an-2 Calculate the third, forth and…

Q: Suppose that the sequence x0, x₁, x2... is defined by xo = 3, x₁ = 6, and xk+2=-6xk+1-5xk for k20.…

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Q: Suppose we have a recursive sequence ƒ1, ƒ2, ƒ3, . . .. For the purposes of this problem, it does…

A: Observe that in the inductive step, to determine fk+1, you need to know two step back fk-1. So, we…

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: 2. Prove the sequence an = n"/n!, n > 1, is %3D (a) increasing; (b) not bounded above (show an 2 n).

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Q: 3. Find the unique sequence (an) of real numbers with Σk-0 akan-k = 1 for all n ≥ 0.

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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: 2. Suppose that the sequence {an} converges to a and that a < 1. Prove that the sequence {(an)"}…

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Q: Find the limit of the sequence {(₁+ 3 n+ 5 2n S

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