Which of the followings are true? I. If , a, is convergent then lim,+∞ an = 0. II. If lim,-+00 an = 0 then 1 an is convergent. III. If 500 n=1 %3D En=1 a, is convergent then 1000 a, is convergent. O a. II, II O b. II O c.I O d. I, II e. I, II

Q: If lim an 1 then n00 Σ an n=1 converges O diverges O may or may not converge

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Q: Σ 2n n? + 2n h= 1

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Q: If 2n=1a, converges, and if a„ # 1 and a,n > 0 for all n, a. Show that E=1a, converges. b. Does…

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Q: +o0 S 13 In n n=1 Determine whether the sequence converges, and if so find its limit. If the…

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Q: Which of the followings are true? I. If 1 an is convergent then lim->00 am = 0. II. If lim-00 n = 0…

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Q: a) Determine if Eº, absolutely converges, conditionally converges, or diverges. ´n=2 n-yn

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

A: Given an &lt; bn &lt; cn

Q: In(n + 9) State whether the sequence ɑn = converges and, if it does, find the limit. n а) converges…

A: Given data is : an=lnn+9n

Q: Suppose (xn) converges, and let K be a positive integer. Now create a sequence (yn) by changing the…

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Q: 2. Define a sequence (a) by a=1, an+1 dn Vn E N. (n + 1)2 Does the sequence converge? Find the limit…

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Q: 2 then E " Cn+1 converges. 3.11 limnr+m

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Q: For each positive integer k, let H, (k) be the smallest harmonic number larger than k and define the…

A: As per the question we have to check that wheather the given sequence E(k) is convergent or not.

Q: Does the sequence {(n^(3)+6n)/(e^(n))} converge or diverge? Why? If it converges, find its limits.

A: The solution is in step 2.

Q: E. Suppose that the S, satisfies both lim,-o0 S20 = -1 and lim,-00 S2n+1 = -1. (That is, the…

A: A sequence an has the limit L if for every ∈&gt;0 there is a corresponding integer N such that…

Q: In the given question as follows, determine whether the statement is true or false. If it is false,…

A: Given statement, If limn→∞ an=0  then ∑n=1∞an converges

Q: 5n+1 converges and, if it does, find the limit. State whether the sequence ɑn = 6"-1 а) converges to…

A: given sequence ,                        an=5n+16n-1

Q: A. Consider the recursively defined sequence s¡ = 1 and Sn+1 = 1 / (7 - Sn) for n 2 1. i. Prove that…

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Q: If lim =c>0 , then b, , a, both convergent and both divergent.

A: By limit ratio test we know that if above condition is true then either both series diverges and…

Q: Label the following statements as true (T) or false (F). lim, 00 (4n² – 5n) = e If lim,-00 an = 0,…

A: (a)

Q: Let a, = vn +1– yn. Then, lim a, = None of the other options -00 and the sequence (a,,} is divergent…

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Q: If lim a, is not 0 then 2 a, is convergent n=1

A: By using theorem we can say that

Q: 3 - If E is convergent, find its sum. z=1 en n(n+3)

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Q: Suppose that (a, } is a monotone sequence such that a, S11 for all n. Must the sequence converge? If…

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Q: Are there any values of x for which E=1(1/nx) converges? Give reasons for your answer.

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Q: Question 15 -(#)" n 11 State whether the sequence ɑn = converges and, if it does, find the limit. а)…

A: Given sequence is an=11nn

Q: 4. Let r be a positive number. Determine the value(s) of r, if any, such that converges. As n" n=1…

A: Explanation of the answer is as follows

Q: 5. (a) Suppose 1 and Xn+1 = 2 Show that {n} converges and find the limit. X1 for all nEN.

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Q: e-n n(n + 1) E(-1)"+1. n=l

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Q: 5n+1 converges and, if it does, find the limit. State whether the sequence ɑn 6"-1 a) O converges to…

A: We have to check whether the sequence converges or diverges. If converges then find the limit.…

Q: Which of the followings are false? 1. If E1 an is convergent then lim,n+00 An = 0. II. If lim,„→00…

A: Series convergent results are given, we have to find the false statement.

Q: E (1- nsin-) converges n = 1

A: Series convergence test problem.....

Q: 9. Determine whether the following is convergent or divergent +(-1)"+1 n! 2n 2e

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Q: +00 El-1)» (" – n)" 2η n=1

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Q: If {A_c} up ∞ down c=1is a convergent sequence , then {1/A_c} up ∞ down c=1 is a divergent sequence.…

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Q: k ) , which statement is true for series =1 Uk = Ek=1 Uk uk+1 2k If ρ -lim

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Q: b) E, 2n-1

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Q: If > an converges, then a converges. n=1 n=1

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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: If 0 1, which of the following must be true? A) If lim an = 0, then > bn converges. в If an…

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Q: ca, is convergent. is an convergent and c is a nonzero constant, then If n=1 n=1

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Q: Suppose 1 aj = 2 3 4 5 aɔ = az = as = 1 1 3 4 4 5 6. 6. a) Find an explicit formula for a,: b)…

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Q: Given that {an}n=1_is convergent such that a1 = V5 and an+1 V5 an- %3D Solve for lim an n 00

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Q: If lim,→00 an 0 then an converges. Select one: O True False

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Q: (b) Show that the following sequence is convergent: 1+ (-1)" n² Yn = √2+ (c) Show that (yn)neN is…

A: A Sequence &lt;an&gt; is convergent if and only if limn→∞ an is finite and unique.

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: A. Consider the recursively defined sequence s1 = 1 and s+1 = 1/ (7 - Sn) for n > 1. i. Prove that…

A: To show sequence sn is convergent, we used the monotone convergence theorem, which says that every…

Q: Suppose 16 1 4 a4 = 1 4 a1 a2 аз a5 1 6. 2 3 5 6. a) Find an explicit formula for an: b) Determine…

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Q: If sequences {an} and {bn} both diverges to infinity, An is convergent. bn then the sequence

A: We have to find

Q: If £a, converges and Eb, diverges, can anything be said about their term-by-term sum E(a, + b„)?…

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Q: 1 3) For which a does converge? Justify your answer. n(ln n)a n=2

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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.