8. 2. Let an be convergent with an > 0 for all n. Now consider (an). n=1 n=1 a) Explain in words why it must be true that (an)2 < an for sufficiently large n. b) Does (an)² converge or diverge? Explain in words and be sure to cite a test n=1

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A: Follow the steps and just remember the form of the nth tems test.

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A: For the given series

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Q: 1. Use an appropriate way to determine whether the following is convergent or divergent. k 'Ink i. Σ…

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Q: I. Consider an = V4n² + 3n – 2n for n >1. 1. Determine if {an}-1 is convergent or divergent. 2. Is >…

A: Simplify the expression of an. Multiply and divide the given expression of an by 4n2+3n+2n to…

Q: la) 을 In ? n=2 16) n=2

A: The solution of given problem is given as follows :

Q: an Q4. If £ an converges and an > 0 for all n , can anything be said about E- ?

A: Given that ∑an converges and an&gt;0 for all n. The objective is to determine whether ∑1an converges…

Q: Supposean converges conditionally (but not absolutely). Let A+ = {n €N : an ≥ 0} n=0 and A_ = {n € N…

A:

Q: 1. Write a paragraph at a level that a high school student would un- derstand that explains your…

A: The objective is to write a paragraph on approximating the irrational number 7.

Q: 7N 8N 3) Show that converges/diverges.

A:

Q: The sequence (n+(-1)" n Diverges Converges to 0 Converges to 1 Converges to e Converges to 2e

A: Option 3 is correct.

Q: 4. Given the FOl owing conditions, give an example and explain your an swer. 8 a. Cn is convergent…

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Q: (3n)" (2n + 50)" n=1

A:

Q: · E In (e + +) n = 1

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Q: From Rogawski ET 2e section 10.3, exercise 3. Use the Integral Test to determine whether the…

A:

Q: 1. Determine whether n² – sin(n) is absolutely convergent, conditionally convergent, or diver n6 +…

A: Solve the following

Q: 'Зп+5 100 8) Use the Root Test to determine convergence or divergence: Ln= n=1 4n+3,

A: solution:-

Q: 10 Show that (S) converges to a continuous function, but not uniformly. Use the series E f, to show…

A: Solution: The function fnx is defined as follows: fnx=0              x&lt;1n+1sin2πx     1n+1≤x≤1n0…

Q: 4. Show that the sequence {fn}, where fn(x) = xER 1+n°x*' is point-wise convergent but is not…

A:

Q: 4. Show that the sequence {fn}, where nx fn(x) = xER 1+n°x is point-wise convergent but is not…

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Q: 2 n 1 o Show that + converges/diverges n-

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Q: 3. Show that the ratio test is inconclusive, but use either the integral or a comparison test to…

A: Given the series:

Q: 34. Assume that an+1/an converges to p = 4. -1 Does > an converge (assume that n=1 An # 0 for all…

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Q: Show that the series n' (2n)! is convergent. Deduce that lim (2n)! n=1

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Q: .8.8 Suppose that Cn >0 for each n E N. Prove that n either converges = diverges to infinity.

A:

Q: Find the smallest positive integer n so that S, approximates the sum of the convergent ser to four…

A: Here we have to find the smallest positive integer n so so the Sn, approximates the sum of the…

Q: 1 3. Use the Limit Comparison Test to determine whether converges or diverges. Vnvn+1 n=1

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Q: Determine if the given is convergent or divergent. Indicate what test is used In n 1. n=1

A: To determine that the following is convergent or divergent  ln2nnn=1∞ ∑n=2∞ln2nn Root test: If…

Q: Suppose that a, 20 and b, 2 0 for alln> N where N is an integer. If an lim = 0, what can you…

A: Consider the given: Limit Comparison test:an,bn&gt;0  ∀ n≥Nlimn→∞anbn&gt;0,∑n=1∞an converges then…

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A: Given: ∫0∞1x1+ln x2dx

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Q: Suppose an is conditionally convergent. Define what this means, and then use a proof by n=1…

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A: We have to prove that if convergence of (sn) implies the convergence of (|sn|).

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Q: The series 3 k 1 3. is 3. convergent to O None divergent convergent to 3 convergent to

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Q: The series 2n +2 Diverges to-o0 Converges to 1- Converges to B Diverges to +

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A: Given problem:-

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Q: In(n) 4. The series is convergent. Show that this is so by using the Integral Test and then again by…

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Q: Does (-1)r+1 In n (-1)n+1 lnn converge or diverge? n=1

A:

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A:

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A: we have to test whether the given series is convergent or divergent by using directly divergent test

Q: 1.4 Does (fa(r)} in 1.3 converges uniformly on |-3,3]? Explain.

A:

Q: (e) $₁ = 5 and Sn+1 = √√4s +1 for n ≥ 1.

A: Given s1 = 5 and sn+1 = 4sn+1 for n≥1

Q: Show that the series n" (2n)! is convergent. Deduce that lim 0. (2n)! n=1

A: