4. Define a sequence by an+1 = -an?, with -1 < ao < 0. Prove that {an} is increasing and bounded above.

Q: Determine the monotonicity and boundedness of the following sequence 4" 2n + 100 an=

A: Determine the monotonicity and boundedness of the following sequence a_{n} = (4 ^ n)/(2 ^ n + 100)

Q: true or false: if the sequence [a sub(n)] is monotone decreasing, then [a sub(n)] must converge

A: Follow next step

Q: 30. а, 2 1+ %3D - n

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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: 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: 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: If the sequence {a}, is convergent, then the sequence {x} defined by a₁+as+as++an-stan is also…

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Q: Exercise 5. Prove that a non-decreasing (resp. non-increasing) sequence which is not bounded above…

A: Let an be a non decreasing sequence . ⇒an&lt;an+1  ∀n∈ℕ Since an is not bounded above so for any…

Q: Show X n² n² +1 is a Cauchy sequence using the definition.

A: We have to check weather the given sequence is cauchy sequence or not.

Q: Prove that if a sequence (an)1 is not bounded, then there is a subsequence (akn)-1 which either…

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Q: 7. let (an)neN be the sequence defined as a, = + (i) Find the limit of an (ii) justify your limit in…

A: Limit of the sequence and and jestification of the sequence.

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: b) With justification give an example of a sequence such that - 1, 1 are its two limit points and no…

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Q: 6. Find the limit of the sequence. You should consider the behaviour of the continuous version of…

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Q: Let (2) be a bounded sequence. Prove that if (en) is monotonically decreasing then it con- verges to…

A: To prove that a bounded, monotonically decreasing sequence {xn​} converges to its infimum, we need…

Q: Prove that sequence {an}= {(-1)n} does not have a limit.

A: According to the given information it is required to show that {an}= {(-1)n} has no limit.

Q: If {b,}-1 is a bounded sequence and lim,→∞ An = lim,-+∞ anbn? Justify. 0, what can you say about

A: Given that  bn is bounded sequence, so, by definition of bounded sequence  ∃B&gt;0 such that bn&lt;B…

Q: 1. Consider the sequence {an} = {-250, 100, -40, 16, 32). (a) Determine whether the sequence {a}…

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Q: A sequence {an} is defined by a1 = 2 and an+1 =3(1+an)/5+anfor all n ∈ N. Find the limit of such a…

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Q: Let {a,} be an infinite sequence with the property that 0 < a, < for all n 2 1. Calculate lim,-00…

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Q: Q1/ Isthe sequence {a,} converge ? († - ) -

A: If a sequence converges to some limit, then it is convergent; otherwise it is divergent.

Q: Is the sequence a, convergent or divergent? Exp n+1

A: The given sequence is

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

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Q: Does the sequence whose nth term () 2 an n converge? If so, find lim,→ an -

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Q: Use the definition to prove that {(-

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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: Show directly from the definition that the sequence {xn}, defined by în Cauchy sequence. = 3n+2 n+1'…

A: Definition of Cauchy sequence xn of real numbers: ∀ε&gt;0 ∃ n∈N  such that |xn-xm|&lt;ε ∀m&gt;n

Q: 1. Let a, = 1 and an+1 = 1+ 1+an (a) Find the first six terms of the sequence {an}. (b) Prove that…

A: As you asked multiple sub parts so I answered 1st three subpart.

Q: Consider the sequence defined by a_n= (4n + (-1)^n2 - 2)/8 for every integer n > 0. Find an…

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Q: Is it true that a sequence {an} of positive numbers must converge if it is bounded from above? Give…

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