Let fn(x) on [0, 1]. 1+ x" (a) Prove that fn converges uniformly to 0 on [0, €], Ve E (0, 1). (b) Does fn converge uniformly on [0, 1]? Prove or disprove.

Q: (b) Prove that the improper integral Ï Ï -8 -∞ converges. dx dy (1 + x² + y2)3/2

A: Consider the given improper integral, I=∫-∞∞∫-∞∞dxdy1+x2+y232 Change into circular coordinates by…

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A: We know that if a sequence of function has bounded derivative the uniformly convergent

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Q: Show that the following method is convergent, and determine it's global order of accuracy. 1 At (Yn…

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Q: Exercise 3. Find f(x) = lim,0 fn(x) on S. Show that (fn)n converges uniformly to f on closed subsets…

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Q: Suppose fn:A→ Ris uniformly continuous on A, Vn E N and fn →f converges uniformly on A. Question:…

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Q: 1) Does cos" cn) Sin*™(n) converges uniturmly on D, : [0, K14] ?. And on

A: Given sequence of function is, fnx=cosnxsin2nx. It can be written as, fnx=cosxsin2xn. If common…

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Q: Suppose that |f(r)|I b where and g(r)dx converges. Prove |f(x)dr converges. that

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Q: Let g : [0, 1] → R be continuous and g(1) = 0. Define gn(x) = g(x)x" for x E [0, 1] and n = 1,2,…

A: As per our guidelines we are supposed to answer only one question.

Q: 1 (a) Use the Integral Test to show that > converges. 2. n' +1 d 1 n = 0 -1 tan dx Hint: 1+x?

A: As per our guidelines, we are supposed to solve only one question.  Kindly post other questions  as…

Q: The improper integral 2 dx (x-1)* X - a) Converges to -2 b) Converges to -5 c) Converges to 0 d)…

A: solve the following

Q: 8. Let f : R → R be uniformly continuous on R and if fn(x) = f(x + ) for x E R, show that (fn)…

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Q: If |fn(x)| < an for all r and E=1 an converges, prove that En=1 fn(1) converges uniformly.

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Q: A sequence of functions fn and function f such that fn is not continuous on 0, 1] for all n, f is…

A: Given below the detailed solution

Q: Let fn (x) uniformly converge to f(x) on [a, b]. If fn(x) E R[a, b], then f(x) E R[a, b] True False

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Q: 10 Detemine whether the improper miegral lo F+16) dx diverges or converges. Evaluate the integral if…

A: Given the improper integral  I=∫0∞10x(x+16)dx

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Q: A sequence of non zero functions fn a nonzero function f both defined on (0, 1) such that fn 1 does…

A: See the attachment.

Q: Consider the series xt = sin(2πU t), t = 1, 2, . . ., where U has a uniform distribution on the…

A: Given that,        Xt= sin(2πU t), where,       t = 1, 2, . . .,       U has a uniform distribution…

Q: Ee whether or not fn converges uniformly nclusion. b) (0, 1)

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Q: Let fn (æ) = a" – x2", defined on [0, 1] %3D is pointwise converges to f (x) = 0, on [0, 1] then ||…

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Q: z . Prove that fn (z) = sin is uniformly convergent to 0 on D(0,2).

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Q: Show that f„(x) = x" (1– x") converges pointwise V x € (-1,1]. Is f„(x) converges uniformly?

A: When a sequence includes a variable, it becomes a sequence of function. A sequence of function can…

Q: Determine whether the improper integral x²e=x*dx converges and, if so, evaluate it. 8.

A: ∫-∞+∞x2e-x3dx

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

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A: Here we use epsilon delta proof

Q: Let fn(x) = x¹(1-x), r [0, 1] and n € N. Show that fn = 0 on [0, 1] but that {f} is not uniformly…

A: Given fn(x) =xn(1-x),             x∈[0,1] and n∈N To prove: {f'n} is not uniformly convergent.

Q: (b) Check the convergence of the improper integral dx. I+2

A: Improper integral

Q: 1 dx et+ e

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Q: 4. Suppose fn:A→ Ris uniformly continuous on A , Vn e N and fn →f converges uniformly on A.…

A: Given: fn: A→ℝ is uniformly continuous on A ∀n∈ℕ To prove: fn→f converges uniformly on A  fn→f…

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Q: Consider the sequence of functions n+ cos(x) 3n + sin(x) defined on the R. 1. Find the function f to…

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Q: 1. For r < 1, define fn(x) = ²x. (a) Prove that for x < 1, k=1 ƒ(x) = lim_ƒn(x) = 84x x(1+x) (1-x)³…

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Q: Let {fn}=1 be a sequence of continous functions on an interval [a, b]. If {fn}n=1 converges…

A: As per the the instruction in the question, we will be solving part (a) - (i). Let fnn=1∞ be a…

Q: 2. Determine the convergence of the improper integral e² In x dx.

A: Take the integral, ∫0∞xlnx=lnxx22-∫0∞1x.x22dx=x2lnx2-x240∞=∞ As after taking x→∞ the limit of the…

Q: Let fn(x) = nxe-n, for x = [0, 1]. (a) Show that for each x € [0, 1] we have fn(x) converges to 0.…

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Q: 1 Determine whether the integral #(lnr)3/2 da converges or diverges. If it converges, give its…

A: I have used limit method  to evaluate

Q: 2. Let 3nr? + cos nr In (r): %3D (a) Find the pointwise limit of (gn) on R. (b) Is the convergence…

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Q: 1 Given f : [2, 0) → R, Prove that fn(x) = converges uniformly to f(x) = 0. 1+ x"

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Q: Suppose that a sequence of differentiable functions {fn} converges pointwise to a function f on an…

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Q: Determine the convergence of the integral tan x d.x. E (In(cos r))2

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Q: If fn : R → [0, 0) is a sequence of integrable functions, fn + f a.e. as n → 0, and S fn + 0. Prove…

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Q: 5. Let (n) be a uniformly bounded sequence of functions which are Riemann integrable on [a, b, and…

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Q: 2. Use the Comparison Theorem to determine whether the integral x +4 J. V-2-1 dx x4 – 2x – 7…

A: Comparison Test for integrals: If fx≥gx≥0 on the interval [ a, ∞), then, 1. If ∫a∞fxdx converges,…

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

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Q: fn (x) = Vx e (1, 00), Vn E N. Explain why the sequence {fn} does not satisfy the conclusion of the…

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