1. Vk +1- Vk = 1 Consider the sequence ak = and the infinite series ak. Which of the following Vk+1+vk k=1 is true? (A) lim ap O and ak converges. k=1 (В) lim ak + 0 and ak converges. k=1 (C) lim ak O and ak diverges. k=1 (D) lim a 7 0 and ak diverges. k=1

Q: Prove the lim(Sn) = -∞ case.

A: We use the notation uN =inf{ sn :n>N},              vN =sup{ sN : n>N }, u = lim uN = lim inf…

Q: For the sequence, determine if the divergence test applies and either state that lim n→∞ an does not…

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Q: (2) Show that each of the following sequences (an) in R is convergent, and determine limn- an: n² +n…

A: According to guidelines we can do only first three sub parts.. thank you A sequence is said to be…

Q: 1 2 + xn (a) Prove by induction that în > 0 for all n € N. (b) Prove that (n) is a contractive…

A: (a) We have to prove by induction that xn > 0 for all nЄ N.(b) We have to prove that (xn) is a…

Q: Let (an)neN be the sequence with a₁ = = 2 and an+1 = √16an 15 for all n = 1, 2, 3, . . .. (a) Show…

A: The given sequence is an+1=16an-15, a1=2. (a) To Show: (an)n∈ℕ is bounded above. (b) To Show:…

Q: 1. For each of the following sequences, (i) Compute lim sup and lim inf. (ii) Determine whether the…

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Q: 7. Let {a,} be the sequence defined by a, = n cos(n7) n+1 Do the following. (a) State the theorem…

A: Answer

Q: (ii) If (sn) is an unbounded decreasing sequence, then lim så -∞

A: Given that {sn} is an unbounded decreasing sequence. To show :- limn→∞sn=-∞

Q: 3. A sequence is given by x₁ = 1 and Xn+1 = √4xn-1 for n EN. (a) Show that än is monotone and…

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Q: Sequences and Series Arithmetic and Geometric Sequences and progressions. 1. Write down the first…

A: nth term given , We have to calculate first 5 terms for each :

Q: (a) (b) Find the limit of the following sequences or show why they diverge. In(5n6) 1 7n-1 +n8 8n³ +…

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Q: E 1. Find the third-degree polynomial P such that two of the zeros are 4 and 1 + i and such that…

A: Since you have posted multiple questions i can do first question as per our company guidelines…

Q: (3) Define a sequence by $1 = 0, and Sn+1 = +¹ for all n € N. 3 (a) Prove by induction that 0 ≤ Sn ≤…

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Q: Study the sequence below using Cauchy sequences. V₁ = Cus (a) + Cus (2a) +----+ Cus (na) 23 2.s with…

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Q: (1 + 1) ". (a) Define the sequence {b} by b₁ = 2, and for n ≥ 2, 1 n! Recall that e= lim 84x bn =…

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: The Fibonacci sequence 1, 1, 2, 3, 5, ... is the sequence (fn)nɛN defined by fi = 1, f2 = 1, and fn…

A: Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: 2. (a) Write the first 5 terms of the sequences with general terms an below and indicate whether…

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Q: Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit.…

A: We have to find out limit

Q: 5 Let an = cos(nn) (1-2). Then one of the following is TRUE for the sequence (an}. A Divergent D lim…

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Q: n-1 (a) n+1 Converges or Diverges: limit (If Converges): (b) Ž"-1 n=1 N+1 Converges or Diverges:…

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