3. Let a₁ = = 1, an+1 = √2an for ne N. Prove that the sequence {an}_1 is convergent and find its limit.

Q: Let x₁ = √2, and x1 Xn+1 = 2 + √√xn, n = 1, 2, 3, .... Prove that the sequence {n} converges, and…

A: Let  and To Find:Prove that the sequence is monotonically increasing and bounded above also shows…

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: 2. Consider the following bounded and increasing sequence: a₁ = √3 a2 V √3+√√3 a3 = √√3+√3+√3 √√3+…

A: A monotonically Increasing sequence &lt;an&gt; which is bounded above converges to its least upper…

Q: Suppose that ho, hị, h2, . is a sequence defined as follows: ho = 1, h1 = 2, h2 = 3, %3D hy =…

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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: 2. an+1 sequence? for an Show that the sequence in Q1(a) is eventually strictly decreasing by…

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Q: Suppose that the sequence x0, x1, x2... is defined by xo=4, x₁ = 6, and xk+2 = 4xk for k20. Find a…

A: Given Data: Let us consider the given data, xk+2 =4xk which can be written as, xk+2 -4xk=0with x0=4…

Q: 1. Prove that the sequence {4+} converges to 4.

A: To know weather two function converge or not take difference of them and put limit to infinite. If…

Q: 5. Consider the sequence {a} defined recursively as : a = 0, a₁ = 1, an = an-1 + an-2. Prove by…

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Q: 7. Find a formula for a, of the sequence { 3, 6, 9, 12, ..}, assuming that the pattern of the first…

A: Given sequence is 3,6,9,12,..... Here, we have a2-a1=6-3=3a3-a2=9-6=3 So, a2-a1=a3-a2 Therefore,…

Q: 1. Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the…

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Q: 23. an n √n² +1

A: We take n common from the root and then apply limit to find the vale of limit.

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

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Q: 5. Prove that every bounded complex sequence has a convergent subsequence

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Q: 3. Prove that the sequence with nth un (a) is monotonic increasing, (b) is bounded above, (c) is…

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Q: Let a 1 and define an+1 = V4an -I for n2 1. Show that the sequence {an} converges and find the…

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Q: Let (an) be an increasing sequence of real numbers bounded from above and (bn) be a decreasing…

A: Given an is an increasing sequence of real numbers bounded above and bn  is a decreasing sequence.…

Q: 6. Consider the sequence defined recursively by 1 2 an+1 = an + N an If ao = 1, does the sequence…

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Q: 7. Does the sequence converge or diverge? If it converges, then find the = 3n-√√9n² - n limit: bn

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Q: Solve all 1) Assume that (xn) is a monotone decreasing and bounded sequence. Prove that (xn) is…

A: is monotone decreasing and bounded sequence.We have to prove that  is convergent.

Q: 2. Let the sequence (n) be recursively defined by x1 = √√2 and xn+1 = √2+xn, n≥1. Show that (xn)…

A: The objective of this question is to prove that the given sequence (xn) converges and to find its…