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

Q: Prove that if the sequence (xn)n is convergent, then its limit is unique.

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Q: 1. Define when a sequence X = (x,) in R is said to converge to x E R. Then prove that the sequence X…

A: The given sequence is: xn=3n+12n+5

Q: Determine the limit of the sequence a, = [In(n + 1)-in(n)] 1 e 1 in 2 1 4 e Divergent N O

A: Topic = Limits

Q: Consider the recursively defined sequence s1 = 1 and sn+1 = 1 / (3-sn) for n  ≥ 1. i. Prove that sn…

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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: Consider the sequences (an) and (bn) given by an= (−1)nn2, and bn= 2n. Decide if the sequence…

A: Convergency for an:we can find limitwe know that (-1)^n is oscillating This value can be either -1…

Q: Prove that if {xn} is a sequence such that 1+ n |æn| < for all n E N, 1+ n2 then the sequence {xn}…

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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: (d) Let x* = 1/2(2*), and what is the limit of x* as k 0? What is the order of converge for x?

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: 2. Determine whether the following sequence converges, and if so, find its limit. (a) {Vn² + 3n -…

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Q: Suppose that (an)n20 is a bounded sequence of real numbers. For each natural number n, define fn(x)…

A: We are given a sequence of functions, where {an} is a bounded sequence. We need to establish the…

Q: Consider the following sequence X, = + +3 for n e N. Let L = lim¬0 Xn. Which of the following…

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Q: Suppose that {a, } is a monotone sequence such that a, 10 or increases- in this case we can't say…

A: We need to choose correct options.

Q: Find all values of x such that converges. n! n=1

A: Given that, series ∑n =1∞  xn2n! Compare the series ∑n =1∞  xn2n! with its standard form ∑n =1∞  an…

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

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: Find all values of x such that ΣΗ converges. n! n=1

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Q: 2. Show that if a > 0, then the sequence (n²a?e¬n*) coverges uniformly on the interval [a, ∞), but…

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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: Consider the sequence n²r(1 – nx) x€ (0, 1/n] =: (x)"f elsewhere . Show it converges uniformly on…

A: Consider the sequence  fnx=n2x1-nx      ,x∈0,1n0                       ,elsewhere The limit function…

Q: Is {3n/(3n-1)} n=1 to infinity, a convergent sequence in R with the usual metric dR(x,y) = abs(x-y)?

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Q: Consider the real sequence {an} given by an = In(n + 1) – In(n+ 2), ne Z>0 = NU {0}. Justify whether…

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Q: For each n ≥ 1, let fn : R → R, fn(x) =nx/(n + x^2) Show that the sequence of functions {fn}n≥1…

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Q: Show that the sequence {fn} where fu(r) = e-n* is uniformly convergent on [a, b), a > 0.

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Q: Let {x„}" and {ym}*=1 be convergent sequences with limits x and y, respectively. Suppose that all…

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Q: 4. Show that the sequence {fn}, where nx fn(x) = xER 1+n°x is point-wise convergent but is not…

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Q: 15. Let a and b be constants with 0 < a < b. Does the sequence {(d" + b")/"} converge? If it does…

A: Given :             The sequence is an + bn1n      Definition of convergent :                  The…

Q: 20

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Q: Let (Xn)n and (Yn)n be two convergent sequences with limits x∗ and y∗. Prove that (a) xn + yn is a…

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

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Q: .8.8 Suppose that Cn >0 for each n E N. Prove that n either converges = diverges to infinity.

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Q: Prove using the ϵ−n0 definition that the sequence Xn=(9−7n)/(8−13n) converges, and find its limit.

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Q: 1. Recall the definition of the limit of a sequence of real numbers (xn) = (x1, X2, X3, . . .) "(Tn)…

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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: 3. Show that if x1 = 1 and if In+1 Van + 2 for n E N, then the sequence (xn) is monotone and…

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Q: Consider the recursively defined sequence s1 = 1 and sn+1 = 1 / (3-sn) for n  ≥1. i. Prove that sn…

A: Consider the given sequence:

Q: Let x1 = 1 and define xk+1 = sqrt(2xk) where k is a natural number. Prove that the sequence {xk} for…

A: Given: x1=1, xk+1=2xk So, x2=2x1=2&gt;x1 x3=2x2=22&gt;x2 On generalizing, we get…

Q: 2. Use the definition to prove that if (xn) is a sequence of real numbers which converges to -3,…

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Q: By examining the limit as n→0, the sequence {2"}D=1 O Diverges O Converges to 0 1 Converges to

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Q: State whether the sequence an = Vn +7 converges and, if it does, find the limit. a) O converges to…

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Q: 1 dx where p> 1 %3D xP

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Q: nd the limit of the sequence a, = In(5n + 7) – In n. If the limit does not exist, type "Diverges" or…

A: Topic- Series

Q: (a) Suppose that f : X → Y is a continuous function and {xn} is a Cauchy sequence in X. Does {…

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Q: Let (X,) be a sequence of real numbers. If for some x e R, (X,) converges to poth 7x + 2 and 2x - 13…

A: which of the following statement is true.

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: Prove that the sequence {sin(nn/3)} does not converge.

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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: 5. Show that the sequence Pn = 1 converges linearly to p 0, and the sequence Pn = 10-2" converges…

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Q: Find the limit of the sequence a, n! 1 1 e e Divergent

A: Answer and explanation is given below..

Q: Find a sequence X, Such that X, does not converge (to any XE R) and Y, does converge to 0 where Y,…

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