Let {a,},{b,} be sequences of real numbers satisfying Ja,|s[b,| for all n 21. Then (a) Za, converges whenever Eb. converges (b) 2an converges absolutely whenever 2bn converges absolutely (c) 2b. converges whenever 2a. converges (d) 2b, converges absolutely whenever 2an converges absolutely If La, is absolutely convergent, then which na1 of the following is NOT true? (a) am→0 as n → ∞ men (b) Ea, sinn is convergent (e) Ž an is divergent (4) Ža; is divergent n=1

Q: 18. Which of these sequences is/are not bounded? I. a, = (-1)", n> 1 А. II only В. I only С. П only.…

A: Option (C) is correct. II only.

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A: see my solution

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Q: 5. The subsequence of (xn) = () is.. a) (1층..) 0) (1.층,급.) b) (1,1,1, ..) a) (1.-) 6. The sequence…

A: Every subsequence has a pattern.  According to guidelince we solve only first question.

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A: Ratio test If we have series ∑ann=1∞ Then we will find the limit l = limn→∞an+1an If l<1 , then…

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A: Proof

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

A:

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A:

Q: 7) The sequence an = 2ln (n+1). In (2n+1) a) Converges to 2 b) Converges to 1 c) Diverges d) None of…

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A: It is given that,  ∑anxn and ∑bnxn are convergent for all x in (-1, 1). Then,  ∑anxn = ∑bnxn in a…

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A: We have to answer on the given sequence.

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A: For a vector space X, over a field F, a normed linear space X,· is referred to as Banach space if…

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A: (a) Given S=1,12,12,23,13,34,...,0,1. The members of S heap or cluster near 0,1 and every…

Q: 3. Prove that if (a, } and (b,} are Cauchy sequences of rational numbers satisfying lim (a, - b, = 0…

A: 3.

Q: 2. Let an be the sequence defined inductively by ai = 2 and an+1 = 1 an + 2 an (a) Prove by…

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

Q: i) Let {n} a sequence of non-negative terms that is monotone decreasing and în → 0. If SnX1 X2 + x3…

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A: To Give: An example of two sequences an and bn such that all of the following are simultaneously…

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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.

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A: Ratio test

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A: Given that, limn→∞a(n)≠0 then we have to find ∑n=1∞a(n) is convergence or divergence or depends.