Suppose that the sequence {an} converges to a   and that a>0. Show that there is an index N such that an >0 for all indices n>=N.

Q: Show that the sequence {b,}, where b, What can you say about any sequence with these two properties?…

A: we say any sequence which  eventually increasing as well as bounded above, it converges. It is also…

Q: Let (Sn) be convergent sequence, and suppose lim s, > a. Prove there exists a number N such that n>…

A: Given sn is a convergent sequence, and suppose limn→∞sn>a. Since sn is a convergent sequence,…

Q: Which of the sequences whose nth terms appear converge, and which diverge? Find the limit of each…

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Q: Prove that the sequence ((-1)")=1 does not converge to any number.

A: Solution :- The given sequence is                          {  an }   =   { (-1)n}  then , we have to…

Q: Show that the sequence defined by 1 a₁ = 1 an is increasing and an < 3 for all n. Deduce that {an}…

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Q: {xn} is a bounded sequence such that all convergent subsequences converge to the same number

A: Every bounded sequence has convergent subsequence.

Q: Determine the convergence or divergence of the sequence with the given nth term. If the sequence…

A: Given sequence,

Q: Determine the convergence or divergence of the sequence with the given nth term. If the sequence…

A:

Q: Determine whether the following sequence converges or diverges. If it converges, find its limit. 40n…

A: One simplest method to calculate whether sequence converge or diverge is calculate sequence at n…

Q: Suppose that the sequence {n} is such that xn 20 for all n eN and that lim(-1)"xn exists. Show that…

A: We will use Triangle inequality.

Q: If (an) is a sequence converging to L, with an 20 for all n, prove that L≥ 0 and that (√an)1…

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Q: Show the first five terms of the sequence. Determine whether the sequence converges, and if so, find…

A: Consider the given sequence:

Q: If the sequence {a}, is convergent, then the sequence {x} defined by a₁+as+as++an-stan is also…

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Q: 3. Suppose that the sequence (a,} is monotone, and that the sequence {a)} converges. Prove that the…

A: We have given sequence {an} that is monotonic  and sequence {an^2}  which is convergent. To prove…

Q: Does the sequence an converge? n(n+1) If so, what does it converge to?

A: We need to verify whether the given sequence converges or not. So,

Q: Which of the sequences whose nth terms appear converge, and which diverge? Find the limit of each…

A: We find the limit of the an th term. limx→∞(an)=limx→∞1+(-1)nn=1+limx→∞(-1)nn=1+0=1

Q: Consider the sequence (an : n greater than/equal to 1) given by an=(-1)n/en for all n.  Is this…

A: The given sequence is a (n) = (–1)n/en. Check the sequence is monotone or not.

Q: Determine whether the following sequence converges or diverges. Inn | In(2n)

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Q: Suppose that the sequence {an} is monotone and that it has a convergent subsequence. Show that {an}…

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Q: In the question given as folows, determine the convergence or divergence of the sequence with the…

A: Given sequence is  an=lnn32n

Q: The sequence a = n is converges to 0 o converges to 8 попe a divergent sequence

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Q: For each of the following, give an example of a sequence with that property and justify why it does…

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Q: State if the sequence {xn}1 define below is convergent or divergent: fix -1< a < 1, and let xn =…

A: The convergence of the series can be checked by applying the limits to ∞. The convergence of the…

Q: Assume that the sequence converges and find its limit. 72 a 1 = 5, a n+ 1- I+ an

A: Given that the sequence converges and the sequence is defined as a1=5, an+1=721+an. Let L be the…

Q: Assume that the sequence converges and find its limit. a1= 3, a n+ 1 = V8 + 2an

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Q: 1. Prove that the sequence {4+} converges to 4.

A: To know weather two function converge or not take difference of them and put limit to infinite. If…

Q: Write out first five terms of the sequences, determine whether, the sequence converges, and if so,…

A: We have to find the limit of the sequence and find first five terms

Q: Is {-(n2)} n=1 to infinity, a convergent sequence in R with the usual metric?

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Q: Given: Cn . Is the sequence {c, , convergent or divergent? tan-in

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Q: n n² + 1

A: It is a series question

Q: 4n + 5 (Ь) n

A: To compute the convergence of the sequence

Q: 1. Use the definition of a convergent sequence to show {7 - converges to 7 where {7- is considered…

A: Convergent Sequence in Metric Space: Let be a metric space and let be a sequence in We say that…

Q: Does the sequence whose nth term () 2 an n converge? If so, find lim,→ an -

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Q: Let ₁1 and In+1 = limit. 1 for n N. Show that this sequence is convergent and find its In + 2

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Q: Suppose the sequence {a,} is convergent, in particular lim,-o an = L. Does this mean that every…

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Q: 1. Prove that the sequence converges to . 2n+1

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Q: Prove that the sequence defined by a1 = −3 and an+1 = 14 (2an-1) converges. Then prove that it…

A: Given a sequence with recursive formula,  a1=-3, an+1=142an-1 To prove that the sequence converges.…

Q: The sequence (tanh n) converges to2

A:

Q: Find the general term of the sequence, starting with n = 1. Determine whether the sequence…

A: leta1=2-8=1+1-1+7a2=3-9=2+1-2+7

Q: Let (a,) be an unbounded sequence and (b„) be a convergent sequence. Prove that the sequence (a,-b,)…

A: In this question, we use the method of contradiction to prove the desired result.

Q: Vn+I-yn 205. n

A: we have to test whether the given series is convergent or divergent

Q: prove that the sequence (1/5^n) is convergent and approaches zero?

A: Series and sequences

Q: converges

A:

Q: For the sequence {a„}, with lim a, does not exist, this means the sequence diverges. Select one:…

A: 1. Let {an}∞n=0={(-1)n}∞n=0 is an oscillating sequence and for this:an=-1; n is odd1; n is…

Q: Let {a,}1 be a convergent sequence defined by a1 = V5 and an+1 = V5a,. Find lim a,.

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Q: 14. Suppose (an) is a bounded sequence such that all of its converging subsequences converge to the…

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Q: Determine whether the sequence converges or diverges. If it converges, find the limit. In(5n)…

A: Find the limit of the sequence, in order check the sequence converges or diverges.

Q: An 1+- n

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Q: The sequence (a,) = 2 converges to -2.

A: Let (an) be a sequence. If limn→∞an = L, where L is finite Then the sequence converges, otherwise 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.