) Find the limit superior and limit inferior of each of the following sequences: (a) X = (xn) = (1+(–1)"), n E N (b) X = (xn) = (cos² n E N (c) X = (xn) = 3+ пEN - (rm) = (T) (-1)"n) (d) X = n+1 3.

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A: Given - x1>1 and xn+1=2-1xn for n≥2 To prove - xn is convergent and also find its limit .

Q: Does the following sequence converge? If so, find its limit. If it diverge cos(−2π/3−6/n^6)

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Q: 11. If {an} and {b„} are sequences of real numbers, lim,0 an = 5, lim, bn = 8, and fn(x) = a, + bnx,…

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Q: How would I find if the sequences an=nπ cos(nπ) and an= (n2)(1/n) are convergent/divergent?

A: To examine the given sequence regarding convergence, that is study the limits as n tends to infinity

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Q: 00 Fn+1 { is convergent, and Show that the sequence Fn n=1 1 Fn+1 lim n→∞ Fn Fn+1 > 1 and Fn-1…

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Q: = 3. Find the pointwise limit g of the sequence {n} on I = [0, 0), where gn(x) = the E-definition…

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Q: Given that lim n2(x- 5)" converges by the Ratio test. If x€ Z then the numerical n- 00 value of x is…

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Q: Let {xn} be a bounded sequence on R. Show that Xn lim = 0. - n→+∞ n

A: follow next step

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A: This is a problem of power series.

Q: B. Prove that limx → c f(x) = o if and only if for every sequence aŋ that is in the domain of f and…

A: Consider a function fx. Suppose that limx→cf(x)=∞. That is for all N>0, there exists a δ>0,…

Q: Use series to evaluate the limits

A:

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A: A sequence is said to be bounded if it is bounded above and below by a number. That is, the sequence…

Q: because 01. > 1. 1. en n-00

A: If x>1 then limn→∞xn does not exists.

Q: 17. Let (Hn) be a sequence defined by Hn k' k=1 1 0, n +1 (b) Deduce that In(n + 1) < Hn < Inn+1…

A: Note: Since we can solve first three subparts of a question, we have solved the first three…

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A: We have to find the value of x such that x, x+2, and x+3 are consecutive terms of a geometric…

Q: 7 1+ Consider the series an where an = n=2 Compute the limit: lim Van (Enter "infinity" or "inf" if…

A: Use the result limn→∞1+xnn=ex.

Q: Apply the ratio test to the series. n! 2n3 n=1 an+1 Find the limit lim an

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Q: Use series to evaluate the limit. -1x2 x - In(1 x) 2. lim x0 +3

A: Known fact:

Q: Theorem 72. (Products of two convergent sequences) Let {px} be a sequence that converges to x and…

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Q: (b) Prove that if a sequence converges, then its limit is unique. That is, prove that if limn→0 Sn =…

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Q: 1.1 Determine whether the sequence (Jn(x)} = {, on I = [0,1]. %3D I7 converges uniformly

A: As per our norms we have to write one question at a time kindly repost remaining questions.

Q: 4. Consider the folowing sequence of functions. 5+ nx? fn(x) = , x E [0, 1] 5+ nx a) Find the…

A: (4.) We will find the all step of the function.

Q: 3.2.6 Theorem If X (x) then a < lim(x) b. is a convergent sequence and if a S X5bfor all n e N,

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Q: 3. Let a sequence (xn) be defined recursively by letting 1 =1 and xn+1 = In for n > 1. 4n2 (a) Show…

A: We are given recurring sequence, x1=1xn=1-14n2xn We need to,  Prove Xn converges Find the limit of…

Q: 1. Consider the sequence Xn  = √n + 1 − √n, n ≥ 1. Prove that (xn)n is convergent. Find its limit.

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Q: • Let (xn) and (Yn) be bounded sequences and xn < Yn. Show that lim inf xn < lim inf Yn . n00

A: Sequence question

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Q: Given that lim n2(x- 5)" converges by the Ratio test. If x EZ then the numerical n- 00 value of x is…

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Q: 2.21, Suppose that {an}n€N is a Cauchy sequence in a metric space X, d suppose there exists a…

A: We need to show xn converges to x

Q: I have no idea how to approach this question

A: Known fact:

Q: п+1 2n +1 dn + Vn² · 6n+1. 2n ||

A: Note some standard results of limit: limn→∞ 1+1nn=e limn→∞ n1n=1 Suppose xn and yn are two…

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Q: Let the sequence {fn(x)} be defined on [0, 1] by n – n2x; if x E (0, ) fn(x) : otherwise SHOW THAT:…

A: The given sequence is fnx=n-n2xif x∈0,1n0otherwise First to find the limit function of this sequence…

Q: Find the limit of the sequence, using L'Hôpital's rule when appropriate. Vn Vn + 5

A: let I be limn→∞nn+5 Recall: limx→afxb=limx→afxb

Q: Determine if the given sequence converges or diverges. If it converges, what is its limit? (a) {: 5…

A: Qa,b,c asked and answered.

Q: 2n – 7 9. Prove that the sequence with nth u, Зп +2 a) is monotonic increasing, b) is bounded above,…

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A: I have proved using Apsilon delta definition

Q: Construct a sequence of continuous functions fn : R →→ R such that 0 < fn S 1, limn-o Jo fndx = 0…

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Q: 11. Let a, b ER with a < b, and let {fr} be a uniformity convergent sequence of continuous real-…

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Q: 1+ 2n² Xn {Tn} 1– 3n2 - Prove that the sequence where is bounded.

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Q: (금) - for n eN = {1,2, 3, ...}. 1. Let an = sin' sin n+1 1. Determine if the sequence {an}=1…

A: The given sequence,  an=sin-11n-sin-11n+1 for n∈ℕ=1,2,3,....

Q: The sequence a=n-vn+1Vn+3 converges to the limit A) -3 B) -1 C) 1 D) +2 E) -2 F) 0

A: For a series ; if it converges then . If  the limit ; then the series diverges.

Q: Given that lim n²(x- 5)" converges by the Ratio test. If x e Z then the numerical n- 00 value of x…

A: Introduction: The ratio test is also used to test the convergence of a sequence. According to the…

Q: Suppose that {xn} is a convergent sequence and {yn} is a sequence such that for any e > 0 there…

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Q: A) If (xn) is a decreasing and bounded sequence. Prove that (x,n) is convergent with lim xn inf {xn…

A: Multipart :    xn  is decreasing and bounded sequence. To show that , xn is convergent and limn→∞ xn…