4. Use the definition to show that {e "a is a Cauchy sequence. n=1 5. Suppose that a₁ = 1 and an = 2an-1+1 5 for n2. Show that {an) is a convergent sequence.

Q: 3. For each part, let {an} be the sequence with the given general term starting v Answer the…

A: As per our guidelines we can answer only 3 subparts of the given problem so kindly repost the…

Q: Find the explicit formula of the following sequence. 8 (72, 24, 8, 3, ...} O a O b Oc - Od ƒ(n) = a₁…

A: Given sequence is 72, 24, 8, 83, ...

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: 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: 14. A basketball tossed vertically upwards reaches a maximum height and then falls to the floor.…

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Q: Define the sequence {xn} by (see photo). Prove, using the epsilon-N definition, that this sequence…

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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: 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: 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: If (an) is a sequence converging to L, prove that (an)=1 converges to L.

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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: s of stick t

A: Given that it needs 5meters of stick to make a picture frame.

Q: Suppose that the sequence {an} is monotone and that it has a convergent subsequence. Show that {an}…

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Q: 1. Let the sequence defined by (5" +7") be given. Show that the sequence is (a) bounded and…

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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: 1. Determine whether the following sequence converges or diverges. If the sequence converges, find…

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Q: Choose the correct answer. 1. The sequence (x) = (-n) is. a) Cauchy b) bounded e) divergent d)…

A: We have1) The sequence (xn)=(-n) is . . . . . . . . .Now we have take n=1 x1=-1put n=2x2=(-2) So, we…

Q: 6. Directly use the definition of the Cauchy sequence to show that: (i) an = 1/n is a Cauchy…

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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: SOLVE BY LINEAR METHOD - y'- 3 X Y = x 3 cos (x)

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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: 13. Let (an)=I be a Cauchy sequence, and let p : N → N be a one-to-one function. Show that the…

A: A sequence is called a Cauchy sequence if as the sequence progresses, the terms of the sequence…

Q: Bonus. Suppose that {a,}n is a convergent sequence and that {b„}n is a divergent sequence. Can the…

A: {an+bn} can't be convergent.

Q: 4. Consider the sequence Onvergent, to what? 3n+7 Determine whether this sequence is convergent or…

A: Sequence and it's convergence

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: 33. Use the Squeeze Theorem to prove (sinn/n} c) {3"/n!} b) fn/2"} f by contradiction. a)

A: 33.) use squeeze theorem to prove that each of the following sequences converges a) sinnn  , (b)…

Q: If a, is an increasing sequence and every term an is less than 100,000 then an converges.

A: we have to find the given equation is true or false.

Q: 5. Does the sequence {{²} (−1)” n diverge? Explain converge or

A: