In which of the following options, the statements given below and the general terms an that satisfy these statements are matched correctly? I: >an is divergent when the sequence {an}- is convergent n=1 II: E: is convergent when an n=1 2 an is divergent n=1 III: E is divergent when an >an is convergent n=1 K: an =n4 L: an M: an 5 n (а) 1-М (b) I-L (c) I- M (d) I-L (e) I-K II - K II - K II - L II - M II - M

Q: 2. Let 0 b, for each n e N. With the fact in part (a), by simple computations (you should try it as…

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Q: Which of the options is correct for the following sequence- {n(-1)* Sn=0 ütfen birini seçin:…

A: Here, the given sequence is: n(-1)n+2n=0∞ So, the terms of the sequence are: a1=1(-1)1+2=-1

Q: An. If > an converges for some large k, then so will n=1 n=k Select one: True False

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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 is a sequence that is' convergent,descending, and has the smallest upper…

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Q: 4. True or False? You do not need to prove your answers. a) A sequence (an) converges to L if and…

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Q: Prove the following, unofficially known as the Awesome Lemma or Awesome Theorem: Suppose {an} is a…

A: Definition of convergence of sequence is used. Actually there are infinitely many points of a…

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 one of the following statements are NOT correct? I. If E an is conditionally convergent, then…

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

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Q: One of the following sequences is divergent S 2n l Ten In=1 3n ° 1)" n² +7. n=2 O None n- -1)" n² +…

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Q: Which of the following statements is false? O A. The sequences {a,} and {bn} must be positive for…

A: Solve the following

Q: (8) Determine whether the following statements are true and give an explanation or a counterexample.…

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

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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: If Σan converges and an > 0 for all n, can anything be said about Σ(1/an)? Give reasons for your…

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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: If an diverges and 0 (-1)"an converges. n=1 B)>(-1)"b, converges. n=1 c) 2(-1)"bn diverges. n=1 D b,…

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

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Q: Which of the following statements is true? a. The sequence {e"}, is bounded, monotonic and…

A: Check each of the following sequences is bounded, monotone, divergent and converges sequence.

Q: 5. Suppose a, is absolutely convergent. Prove b, is absolutely convergent if |b,| 1.

A: This question is related to series comparison test.

Q: Suppose that a, > 0 for all n and that an diverges. Suppose that b, is an arbitrary sequence of…

A: We have to solve this question with the help of comparison test:

Q: If E=1 an converges, then 5 (-3) an converges. n=5 O True O False

A: We are given two statements related to series and we have to state them as true or false.

Q: IN. 13. Consider the following sequence: a. Determine whether the sequence {an} is monotonic. If so,…

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Q: Which of the following statements is true? O A. Η Σα, converges then Σ al converges. O B. В. Η Σα,…

A: To select: which statement about convergence is true.

Q: If (Xn) is a monotone sequence, then (x,) converges O False O True

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

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Q: Consider the following sequence: {a,) = {n – n²} Sn=1 Use an+1 - An to show that the given sequence…

A: Calculus   sequence

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: Let an 20 for all n and suppose Σan converges. Prove that √an ΣΥ n converges. Hints: One way to do…

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

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Q: A sequence {an} is said to converge to a real number L if for every e > 0 there exists a positive…

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Q: 01: Which of the sequences converge, and which dive 1-6n 2. an an =(1+ 3n %3D an = 1+ 2n ملف

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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: 5. The sequence r. is convergent in (R. d). a. True b. False

A: 5 - b , (false)

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: 3. Prove each of the following. (a) If a converges, then Ei, must diverge. 1+ ak (b) If an 20 for…

A: Idea: If series is convergent then its corresponding sequence is converges to 0. Other ideas we are…

Q: Which of the following statements is true? n 100 a. The sequence {e"}, is bounded, monotonic and…

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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: п2-2n+1 a. an = %3D п-1 (1 -5 n+1 b. аn 2n n'

A:

Q: Do the following: (a) Give an example of two divergent sequences (an) and (bn) for which (an + bn)…

A: Solution

Q: Suppose {an} and {bn} are two sequences. Which of the following statement(s) must be true? O If {an}…

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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: If the sequence (s_n) converges, then so does every subsequence of (s_n). True False

A: Suppose s_n be a convergent sequence. We have to check the statement that every subsequence of s_n…

Q: Let {an} be a positive monotonic sequence with sup an = 12345. Which of the following statements is…

A: Counter example: For option E:  {an}=12345/n for all n€Natural no.  This is positive monotonic and…

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

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Q: 6. Which one of the following statements is true for the convergent sequence (a )? 5(A) Every…

A: Sequence and it's convergence

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.