Prove that the limit of the sequence V2, y2v2, V 21 V2, \/ 2\/2/: /2/2, exists, and find it

Q: Let {an}=1 be a convergent sequence such that a1 = V5 and an+1 = V5 a,n. Compute lim an.

A: The sequence an is convergenta1=5 and an+1=5 an

Q: Give an example of a bounded sequence {a„}_1 having exactly 2 limitpoints.

A: we have to find a sequence such that it has two limit points.

Q: Let x, = 3. Using the algebra of limits and standard results on limits to establish/justify if the…

A: Consider a sequence an. Then if the limit limn→∞an exist then the sequence converges otherwise the…

Q: (Зп + 1)! Find the limit of the sequence an = If the (Зп — 1)! limit does not exist, type "Diverges"…

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Q: 5n 9. Use the L'Hopital's rule to determine if the sequence an = () converges or diverges as limit n…

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Q: . Suppose real sequences (S,) and (t,) are bounded. (That is, that their ranges are bounded sets.)…

A: Given the real sequences sn and tn are bounded. We have to show that The sequence given by sn + tn…

Q: Assume that the recursively defined sequence converges and find its limit. 15 a, = 1, an + 1-2+ an…

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Q: Prove that a sequence in ℝ can have at most one limit

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Q: Determine whether the sequence 2ñ sin converges or diverges. If it con its limit. 27 1 1 2 diverges

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Q: 22: Establish the divergence of the sequence x, =(-1)"

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Q: Compute the DFT for the sequence [1,1,1,1] x(0)= x(1)= x(2)= x(3)=

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Q: determine the limit of the sequence or show that the sequence diverges.

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Q: Prove the sequence converges and find its limit.

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Q: Show that the central limit theorem holds good for the sequence {X;}, if 1 P{X, =± k"} = xk2a, P{X,…

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Q: Suppose real sequences (sn) and (tn) are bounded. (That is, that their ranges are bounded sets.) i.…

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Q: Find the sum from k=1 to infinity of: [5/(3k+1)] using the limit comparison test.

A: Limit comparison Test:

Q: C) Prove that a Cauchy sequence of real numbers is bounded.

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Q: Show that the sequence (x,) = () convergence.

A: We have to use limit n tends to infinity.

Q: 2) Show that the sequence defined recursively by 3 4 An aj = 5 аn+1 satisfies 2 < an < 5 and is…

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Q: If a sequence an is strictly increasing and not bounded above, then the limit of sequence does not…

A: To state whether true or false.

Q: Show that the sequence of functions {f.} where f,(x)=: 1+ nx „xER is non-uniformly convergent on…

A: Show that the sequence of functions fn(x), where  fnx=x1+nx2,  x∈ℝ is non-uniformly convergent on…

Q: 5xn X1 = 1; xn+1 = 1; Χn+1

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

A: 3.

Q: Show that the sequence of functions on the domain [-1, 1] defined by if -1<r<0, 0, if 0 <x < 1/n, 1,…

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Q: Consider the iteratively defined sequence (x,), where x41 = 3 - 2 and x = 3. Use the Monotone…

A: Monotone convergence theorem: If a sequence monotone and bounded then it is convergent.   If the…

Q: Prove or disprove that if a sequence {x,} satisfy x,+1 – xn → 0 then it is Cauchy

A: We have to solve given problem:

Q: Show from first principles (in particular not using any limit laws) that if a sequence (an)ı…

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Q: Show directly from the definition that if (x„) and (yn) are Cauchy sequences, then (X, + yn) and…

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Q: Suppose real sequences (sn) and (tn) are bounded. (That is, that their ranges are bounded sets.) i.…

A: Since sn and tn is bounded then there exist some finite reals Us,Ls,Ut and Lt such that:  Ls≤sn≤Us…

Q: Let {xn} be a bounded sequence of real numbers. Prove that {xn} converges if and only if lim inf xn…

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Q: use the appropriate limit laws and theorems to determine the limit of the sequence orshow that it…

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Q: Prove that the sequence is monotone and bounded. Then find its limit. s1 = 5 and sn+1 = V4sn +1 for…

A: See the attachment

Q: Let {an}=1 be a convergent sequence such that a1 = V5 and an+1 V5 an. Compute lim an.

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Q: Use Theorem 1 to determine the limit of the sequence or type DIV if the sequence diverges. an 49n2…

A: Givenan=2n49n2+5

Q: Prove that the limit of the convergent sequence is unique.

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Q: 28. yn = ne/n

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Q: Exercise 2. Use the definition of limit to prove that both of the sequences {+} and {"} converges to…

A: NOTE: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Show that the limit of a convergent sequence in a metric space is unique

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Q: Use the definition of the limit of a sequence to prove that 3n +1 lim 2n + 5 3 %3D 21

A: We need to use the definition of the limit of a sequence. we have the definition of limit as,…

Q: (Xn) be a sequence of real numbers defined recursively by 1 = 1 and 1 7. Let X = In+1 =5(Tn +3) for…

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Q: If x1, y1, are two positive unequal numbers and xn = (xn-1+ yn-1)/2 and yn= sqrt(xn-1yn-1)  for all…

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Q: 4. Suppose that lim r, = 0. If (yn) is a bounded sequence, prove that lim In Yn = 0.

A: Bounded sequence: If there is a positive number M such that |Xn|≤M for every n∈N, the sequence xn is…

Q: Determine the limit of the sequence or show that the sequence diverges. If it converges, find its…

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Q: Use Theorem 1 to determine the limit of the sequence or type DIV if the sequence diverges. 7n An…

A: To find:

Q: Prove that a sequence of positive real numbers {r„}1 tends to 0 if and only if the sequence {},…

A: We have to prove a sequence of positive real numbers xnn=1∞ tends to 0 ⇔ 1xnn=1∞ tends to ∞.

Q: 8) Show that the sequence 12, V12 + vi2, v12 Is bounded and monotonic. Then find its limit. 12 V12…

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Q: Use the definition of the limit of a sequence to prove that lim (Vn + 2021 – vn) = 0. that jat- %3D

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Q: Show that the sequence {xn}: cost -dt is a Cauchy sequence. t2

A: The given sequence is xn=∫1ncostt2dt. Cauchy sequence:  A sequence xn of real numbers is said to be…

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.