Explore the inconclusive L = 1 case from the Ratio Test! An+1 (a) Define a sequence {an} such that lim An 1 and the series E an is absolutely convergent. Justify all your claims. bn+1 (b) Define a sequence {bn} such that lim bn 1 and the series an diverges. || Justify all your claims.

Q: Given the nth term of a sequence : 1 + + 16 1 1 1 + 2n2 + 4n an 30 (i) Write down a, in sigma…

A: Please refer the attached image for complete solution. THANK YOU.

Q: Prove that if the sequence {an} is divergent and the sequence {bn} is convergent, then the sequence…

A: Given: The objective is to prove that if the sequence {an} is divergent and the seqence {bn} is…

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: True or false: The sequence True False {FL √√n=1 converges.

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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: Use Root Test to determine whether the series E(-1)"en² sin²n n=1 absolutely converges,…

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Q: Assume that the sequence shown below converges and find its limit. √3, √3+√3,√√3+√3+√3 √3+₁ " 3 +…

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Q: Use the Alternating Series Test to determine whether the alte S(-1)k+1_ k 7k + 8 00 k k = 1 Identify…

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Q: 4n2-11 Q2. Determine if the sequence converges, and if so find its limi 12n2+4n-8.

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Q: Let > an be an infinite series. Choose the correct statements below. O If >, am converges, then >,…

A: The objective is to choose the correct statements.

Q: gence of a sequence, prove that the following statements are true. (a) lim 2n+1 5n+4 nx (b) lim 2²…

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Q: Let {an}1 be a sequence such that there exist numbers a and N such that, for n > N, an = a. Prove…

A: We know that the sequence an converges to α when for any ε>0 there exists K such that for n>K,…

Q: Prove that each of the following sequences diverges: (a) {4n - 5} (b) {√3n+ 1} 2n + 3 √n n² +7 8n +3…

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Q: Let (x,) be a convergent sequence of real numbers and let (a,) and (b,) be subsequences of (x,) with…

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Q: (a) what can be said about the series Ea?? Give a reason for your answer. 1+an (b) what can be said…

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Q: Solve all 1) Assume that (xn) is a monotone decreasing and bounded sequence. Prove that (xn) is…

A: is monotone decreasing and bounded sequence.We have to prove that  is convergent.