2. Let the sequence (xn) be recursively defined by x₁ = √√2 and Xn+1 = √2+xn, n≥1. Show that (xn) converges and evaluate its limit.

Q: For each sequence {an}1, prove it converges by showing that it is bounded and mono- tone, and find…

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Q: Q4] Show that the sequence (xn) = (T") for all n EN is not a Cauchy .sequence

A: A sequence {Xn}of real numbers is called a Cauchy sequence if for every positive real number ε,…

Q: 1 25. az = 1, an+1 =(a, +4) a) Since the sequence is recursive find an b) Find lim a,n n- DO c) If…

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Q: Find the limit of the sequence whose nth term is an = In(2n + 1) or show the sequence diverges. Be…

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Q: (a) Give an example of a bounded sequence that diverges.

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Q: Prove that the sequence ((-1)")=1 does not converge to any number.

A: Solution :- The given sequence is                          {  an }   =   { (-1)n}  then , we have to…

Q: Find the limit of the sequences with general term as given: (e) (√4 - √1/n  - 2)n (f) (-1)n √n / n…

A: Answer

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: 5n 3) Determine if the sequence a, = converges or diverges. If it converges, find the limit 2n +1…

A: Explanation of the answer is as follows

Q: 10. f(1) = –0.5 and f(n) = f(n – 1) – 0.5 for n>2 %3D - -

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Q: an = (-1)" 2√n

A: As we know ⇒-an≤an≤anThen limn→∞-an=-limn→∞an=0We have limn→∞-an=-limn→∞an=0 then by  squeeze…

Q: {   x_(n+1) = 3 + [6/(3 + x_n)]     } Beginning with a firs approximation of x_1 = 4, find the next…

A: This is a problem of Sequence.

Q: 2. Using the definition of a Cauchy sequence, prove that (1/n2) is a Cauchy sequence.

A: We have to prove that given sequence is a  Cauchy- sequence , so by definition of Cauchy sequence-

Q: use the theorem in the picture provided to prove that sequence in the other picture converges to 0,…

A: ### Part (a)To apply Theorem 3.2.11 to show that the sequence (nnn!​) converges to 0, we need to…

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: (a,) be a Cauchy sequence of nonzero real numbers. Which of the following statements about (a,) may…

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Q: (-9)"+1 Let an 10n Determine whether the sequence {an} is convergent or divergent. If it is…

A: an=(-9)n+110n A sequence is convergent if value of limit is define and finite as n approaches to…

Q: Let the sequence 9n be recursively defined by go = 0, 91 = 1, 9n = 9n-1 +29n-2 for n ≥ 2. Show that…

A: Given: The recurrence relation, gn=gn-1+2gn-2, g0=0, g1=1 for n≥2 To show: limn→∞gn=∞

Q: 4. Let a₁ = 1 and an+1 = √8+ an Show that the sequence {an}1 is convergent and find its limit.

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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: Is the sequence (N SIN N) properly divergent Is the limit +∞ or -∞0 ?

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Q: 6. Every nondecreasing sequence of real numbers has a limit.

A: Nondecreasing sequence: A sequence &lt;an&gt; of real number is said to be nondecreasing if it…

Q: Assume that the recursively defined sequence converges and find its limit. a, = - 12, an - 1 = /24 +…

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

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Q: Use Squeeze Theorem to find the limit of the sequence {4n/n!}

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Q: Suppose (xn) is Cauchy and (yn) is bounded and monotone. Can we say anything about whether the…

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Q: i need the answer quickly

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Q: can you please show step by step on how to solve this especilaly the induction step so i can…

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