Let do = 1, a1 = -1, a; = (-1)" for 2"

Q: 1. Let z, be the sequence of approximations to the root of f(x) = 0 and r, is a suitably close…

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Q: Q2 Prove that the scries 2+ x2 3+X2 4+x is Unitormly convergem? in any interval?

A: The given series is: 11+x2+12+x2+13+x2+……. Therefore the given series can be written as:…

Q: Show that the series Ln=l 12+2z10n-5z"ª –3z+2;10n converges absolutely for |리 < 1.

A: The series ∑n=1∞an is absolutely convergent if the series ∑n=1∞an  Ratio test:  The series ∑n=1∞an…

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

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Q: Find the interval of convergence for 2 (k + 5) 2* а) O[-3, 3] b) O[-1, 1] c) O (-2,2] d) O[-2, 2) e)…

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Q: 5. Let ni < n2 < n3... be positive integers, and let E = x E [0, 27]; sup converges. Show that E has…

A: Given :  n1≤n2≤n3 ...  be the positive integers and E be the set defined as ;                E =…

Q: Consider the sequence of functions {fn}=2 where fn: [a, b] → R defined for n 2 2 by if 0<x s -n² (x…

A: Definition: Suppose fn is a sequence of functions fn:S→ℝ and f:S→ℝ. Then fn→f pointwise on S if…

Q: 5. Let {fn} be a sequence of measurable functions on E = [0, 1] satisfying Prove that the sequence…

A: Recall.   1: Triangle inequality for integrals.   2: if (a_n)^2 goes to 0 as n goes to infinity then…

Q: Supposean converges conditionally (but not absolutely). Let A+ = {n €N : an ≥ 0} n=0 and A_ = {n € N…

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Q: Find the interval of convergence for Σ (k + 2) æ(k+2) k + 3 а) O1-2, 2) b) О (-1,1) с) O1-1, 1) d) O…

A: Put k+2 = t  We get  ∑(t xt )/(t+1)

Q: 3.3 Prove that does not converge uniformly on R. k! k=0

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Q: What is the largest positive constant K such that if 0 <r < K, then (nl)2 must (2n)! n=1 converge?

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Q: Find the interval of convergence for 2 (-1)*3*x* gk+1 1 1 a) o[- а) b) O(-9,9) c) O[- 3,3] d)…

A: Option E will be the right answer.

Q: Show that if an < o0, then E In(1+an) converges if and only if an converges. n=1 n=1 n=1 Hint: show…

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Q: Suppose that we observe that X1, X2, . . . , Xn  are iid∼ U(0, 1). Show that X(1) converges in…

A: A sequence of random variables Xn is said to converge in probability, denoted as Xn→pX, if…

Q: Show that E-2 ((In n)ª/n²) converges for -o 1. n=2

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Q: Suppose that dz converges. What can we conclude about the constant p? Op>1 Op Op< 3

A: Explanation of the answer is as follows

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: Prove that the seugences or series of functions converges pointwise but not uniformly on [0,1] 1.…

A: We have to show that the following sequence or series of functions converges pointwise but not…

Q: Let Xn be a Poisson(n) r.v. for n = 1, 2, . . .. Prove that Xn/n −→ 1 (converge in probability)

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Q: ht A: If S 2n – 2n²x x € [0, 1] fx,(æ) = { otherwise converges in distribution as n → 0. nt B: If 1…

A: Given that Xn : n≥1 is an infinite sequence of random variables and for each n, the density function…

Q: If 2 a, converges, then la,| diverges. n=1 n=1 True False

A: Given statement is FALSE. I HAVE given a counter -example.

Q: Suppose xo = 2√3, yo = 3, 2xn-14-1 Xn-1+Yn-1 and Prove that (a) xnx and yn ↑y as n→ co for some x, y…

A: The statement of monotone convergence theorem is if a sequence is monotonically increasing and…

Q: Find the interval of convergence of E(-1)". n 2n 36" n=0

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Q: Exercise 2. Determine whether the sequence of functions (fn)n converges uniformly on S. fn(x) = nx"…

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Q: Q3: Show that 3(2)n+2 3(2)n+2 is convergence and En=0 5n Zn=0 5n 20.

A: Please refer to the image below.

Q: Two sequences of functions fn and gn that converge uniformly but fn In does not converge uniformly.

A: See the attachment

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: Suppose xo = 2√3, yo = 3, 4 Note: If (x) is monotonically decreasing (resp. montonically increasing)…

A: As per the question we are given two interlinked recurrence relations between sequences xn &amp; yn…

Q: The series k=1 5( − 1)k+1 - kl.1 O converges diverges.

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Q: The sequence {(1 + 1)^² +1} 2n O O converges to e² is divergent ez converges to e 2 O converges to e

A: Sequence

Q: Let (Jn)neN be a sequence of functions f: R → R satisfying |fn+1(c) – fn(c)| 2 and neN , and ae(0,…

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Q: If fn If 0 ≤ an ≤ bn for all n € N, and f on [a, b], then lim ſå fn ‡ få f. an converges, then bn…

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Q: (4) Suppose xo = 2√√3, yo = 3, 2xn-19n-1 Xn-1+Yn-1 and Prove that (a) xnx and yn ↑y as n→ ∞o for…

A: Hints:  It is given that, x0=23, y0=3 and xn=2xn-1yn-1xn-1+yn-1 and yn=xnyn-1 for all n∈ℕ.…

Q: For n e N let fn: [0, 1] → R be the function fn(x) = e²-n. Decide whether the sequence (fn)converges…

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Q: 1. Show that fn(x) →0 for each x E [0,1]. -> 2. Determine whether or not the convergence is uniform.

A: In this question, we find the limit of the sequence of function and in the second part, we use Mn-…

Q: 2.62 Show that fn(x) = x+n converges uniformly on [0,1] but not on (0, 0).

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Q: Consider the sequence S n²r(1 – nx) x € [0, 1/n] - fn(x) := 0. elsewhere . Show it converges…

A: Let gx=n2x1-nx Differentiating wrt x, we get  g'x=n2-2n3x Putting g'x=0, we get  n2-2n3x=0⇒x=12n

Q: Given two sequences (fn)neN, (9n) nEN C C[0, 1] of continuous functions on the closed unit interval…

A: Given two sequences fn, gn contained in the interval 0,1 of continuous function on 0,1 by…

Q: Suppose {fn}n=1 converges uniformly on E. Show {fn}n=1 converges pointwise on E.

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Q: 3. Let {fn}n=1 and {gn}-1 converges uniformly on E. Prove that {fn - In}n=1 converge uniformly on E.

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Q: 2.65 Let fn(x) = xe¬nx² a) Find || fn||sup for all n. b) Determine whether or not fn converges…

A: The given function is fn(x)=xe-nx2 ∀x∈ℝ To find: (a) ∥fn(x)∥Sup ∀n             (b) Whether…

Q: (a) Prove that 4280 k!pk k=0 converges for any r > 0.

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Q: If E, 2"an converges and Eo(-1)"2"an diverges, then the radius of conver- gence of Eo anx" is 2. =0

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Q: Suppose xo = 2√3, yo = 3, Prove that Xn= 2xn-14-1 Xn-1+Yn-1 and Yn √n Yn-1 for all n € N. (b) x=y…

A: We claim that, xn&gt;yn&gt;0 We shall prove this by principle of Induction. We are given that…

Q: The sequence {G1)}m1 5n+3 5n+1 n=1 Diverges converges to Je3 converges to Je² converges to 0…

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Q: If |Xn+2 − Xn+1| < |¤n+1 − xn| for all n E N then (xn) converges. -

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Q: 2. Let gn(x) to zero uniformly on [a, o). Show that the convergence is not uniform on [0, 00). for n…

A: Given function is gn(x)=nx1+(1+n2x2) Now limn→∞gn(x)=g(x)=limn→∞nx(1+n2x2)…

Q: 7. Let x₁ = a > 0 and Xn+1 = x + 1/x,, for n E N. Determine whether (x,) converges or diverges.

A: Q7 asked and answered.

Q: Consider the sequence of functions {fn}n=2 where fn: [a, b] → R defined for n > 2 by if 0 sxs fa =…

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