In which of the following options, the statements given below and the general terms a, that satisfy these statements are matched correctly? an is divergent when the sequence {an}- is convergent n=1 is convergent when > an is divergent an n=1 II : n=1 III: E is divergent when > an is convergent an n=1 n=1 4 K: an =n М: аn L: an 5 n n (a) I- M (b) I-L (c) I- M (d) I-L (e) I-K II - K II - K II - L II - M II - M III-L Ш-М III- K III - K III - L M³ IM:

Q: 3. n-3n+5 n7 Sn=1

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Q: If an converges and > 0 for all n , can anything be said about E–? An an

A: Given that, ∑ an  converges We known that, the series if  ∑n = 1∞ an  is convergent then limn→∞ an =…

Q: E Suppose we also know that 1 > an > 0. and a converge. Then the series > converge 1-an Select one:…

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Q: (a) Use the Bernoulli inequality to show that for all n ∈ N: 1 ≤ n1/n ≤ (1+√n)2/n ≤ (1 + 1/√n)2 b)…

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Q: Which of the following statements is /are not true? I: Every convergent sequence is Cauchy. II:…

A: A sequence is called a Cauchy sequence if the terms of the sequence eventually all become…

Q: Which of the following assertions is true for the given sequences? a) {(-1)" e"} converges п+1 b)…

A: If you like the solution then please give it a thumbs up.... The Answer is:           Result :…

Q: Which of the following is true about the sequence In (3n -n²+1) - ln(n³ + 2n¹ +5n)? = an 3 The…

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Q: Which of the following is true about the sequence In (3n² − n² + 1) − ln(n³ + 2n¹ + 5n)? = an The…

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Q: 21. Show that E(-1)"+1; 2n converges n! absolutely. n=1

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Q: The series > (2)k-1 converges. k=1 Select one: True O False

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Q: ) Suppose that {am}n20 diverges and {bn}n20 converges. Then the sequence {am · bn}n20 diverges. 7.…

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Q: Q22. A Sequence has been defined as the following 7an a1 = 1, An+1 2+ an You must find its limit,…

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Q: 1. Determine whether the following sequences converge or diverge. If the sequence diverges, state…

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Q: Show that E-2 ((In n)ª/n²) converges for -o 1. n=2

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Q: Which of the following statements is equivalent to Theorem 1? 8. A.lf a, diverges, then a, also…

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Q: 3. Prove the statement: If the series > an is convergent, then a, →0 as n→ 0. Is it also true that…

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Q: 1 1 The series Σ ( – 1)" • Σ( - ( + ) " / n n n=1 converges to ∞. converges conditionally. O…

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Q: Suppose (an) is Cauchy and that for every k ∈ N, the interval (−1/k, 1/k) contains at least one term…

A: Given, an is Cauchy sequence, Let, ank∈-1k,1k ,∀k∈ℕ Then, ank1k , ∀k∈ℕ Let, ε>0 be arbitrary and…

Q: of the following statements is an attempt to snow that a given series is convergent or divergent…

A: some of the statements are given, we have to find its convergence using comparison test

Q: 9. If the series 3 converges, then r=1 A k> -1 B k>0 C k> 2 D k> 3

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Q: Give an example of each of the following, or state that such arequest is impossible by referencing…

A: Let xn=n and yn=-n Both xn and yn are unbounded and therefore diverge However, xn+ yn=n-n=0 Hence…

Q: In which of the following options, the statements given below and the general terms an that satisfy…

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Q: In which of the following options, the statements given below and the general terms an that satisfy…

A: I-M II-L III-K

Q: (4) Let{an}i be a positive sequence which satisfies na,»1 s a, for all n 2 1. an (a) Prove that-1…

A: Given- Let ann=1∞ be a positive sequence which satisfies nan+1 ≤ an for all n ≥ 1.  To find-  Prove…

Q: In which of the following options, the statements given below and the general terms an that satisfy…

A: P-series test:  ∑an=∑1np  P-series is convergent for p>1 and diverges for p≤1. 1) ∑an is…

Q: a) A sequence (an) converges to L if and only if, for every k ∈ N, there exists M ∈ N such that if…

A: A sequence is said to be convergent if it approaches some limit

Q: 8. 5n The sequence 1+ 5n n=1 Converges to Ve O None 1 Converges to 1. Converges to Ve Diverges

A: We have to solve

Q: 1. The sequence {V(n + 4)(n + 6) – n} is A) Convergent to 0. B) Divergent. C) limn-o V(n + 4)(n + 6)…

A: the sequence given is (n+4)(n+6)-n

Q: In which of the following options, the statements given below and the general terms an that satisfy…

A: The divergence test states that if limn→∞an≠0 then the series ∑an will diverge. A sequence an is…

Q: If 2 an converges for some large k, then so will ) n=k n=1 Select one: O True 17 O False

A: It can be solved using the simple observation.

Q: 103n2 – 8 - 1. Xn := 4n2 + 99n + 3 1i. Xn := -n + Vn² + 3n 3n? + 2n iii. Xn := 2n n=1 8.

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Q: Determine which of the following statement(s) will always be true. It is possible that more than one…

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Q: (4) Suppose xo = 2√3, yo = 3, Prove that 2xn-14-1 Xn-1+Yn-1 and Yn √n Yn-1 for all n € N. = (b) x =…

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Q: 1.5 Show that the series (a – 6)(a – 26) „3 + (a- b)(a- 26)(a- 35) | z+- 2! 3! 4! converges…

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Q: In which of the following options, the statements given below and the general terms a, that satisfy…

A: Consider the given information: Here to match the statements and given general term an. Here option…

Q: an If an > 0, bn =n=2 (n ± 0), and lim = 1, then: n→o bn We can apply the limit comparison test…

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Q: The series (k+1)* is k=1 p°k! a) Converges absolutely b) Conditionally convergent c) Diverges d)…

A: Given : series ∑k=1∞k+1kpkk! To determine the convergence of the series and choose correct option.…

Q: In which of the following options, the statements given below and the general terms an that satisfy…

A: P-test : Consider series ∑an=∑1np P-series converges only if p>1 and diverges for p≤1. 1) Given…

Q: If f {an} is a convergent sequence with lim an=0, n+ 00 then E=1ªn is a convergent series. True O…

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Q: In which of the following options, the statements given below and the gener terms an that satisfy…

A: We know that ∑1np is convergent for p>1 and divergent for p≤1   Consider K: an = n2{an} =…

Q: a) Suppose (an) is Cauchy and that for every k∈N, the interval (−1/k,1/k) contains atleast one term…

A: Hello. Since your question has multiple parts, we will solve the first part for you. If you want…

Q: Given a positive sequence {a,}n>o satisfying Vai > yao + 1 ana a < 1 An+1 an-1 for any positive…

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Q: In which of the following options, the statements given below and the general terms an that satisfy…

A: We consider each of the cases L,M,N and match them with I , II, and III

Q: If a <b <c If lim a =L and lim c =L+7, then which of these is TRUE? n- 00 n- 00 OA (b can either…

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Q: If > a, converges but |an] diverges then we can be sure that n=1 n=1 (a) an converges absolutely.…

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Q: Given the expression an = V4n² + 3n – 2n for n > 1 Check whether this is convergent or divergent: an…

A: The given sequence is a an=Sqrt(4n2+3n) -2n.

Q: In which of the following options, the statements given below and the general terms an that satisfy…

A: proof :  Taking k : an = n2              ∑n=1∞an=∑n=1∞1n2 limn→∞1n2=1∞2 (taking limit)=1∞=0  Thus,…

Q: Evaluate the following and detemine if the sequence {a,} converges or diverges 5n2 + 9n 11 12n an 2n…

A: limn→∞an=limn→∞12n3-5n2+9n-112n3-3n2+7n+5=limn→∞n312-5n+9n2-11n3n32-3n+7n2+5n3=122=6

Q: In which of the following options, the statements given below and the general terms a, that satisfy…

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Q: True or False? You do not need to prove your answers. a) A sequence (an) converges to L if and only…

A: We will prove it

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.