For n e N let fn: [0, 1] → R be the function fn(x) = e²-n. Decide whether the sequence (fn)converges pointwise on [0, 1], and decided whether it converges uniformly on [0, 1].

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: In the usual notation, state the Dirichlet test for a functional sequence. Hence show that each of…

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Q: Suppose f: R→R is a continuous, 2n – periodic function. Define the function f,(x) := f(x+ ±) for x E…

A: Given Information -  f is a continuous function on R and is 2π periodic.

Q: the series E k converges uniformly on D(0;r) for every r E (0,1), but diverges on C\ D(0; 1). More…

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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: 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: (x + n)² Show that h„(x) converges poimtwise to f(x) = 1, V x e [1, 00). %3| x² + n²

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Q: Suppose fn : S → ℝ are functions that converge uniformly to f : S → ℝ. Suppose A ⊂ S.  Show that…

A: Suppose fn : S → ℝ are functions that converge uniformly to f : S → ℝ. Suppose A ⊂ S.  Show that the…

Q: Suppose that the sequence of functions {fn} and the function f are integrable on [a, b] and that…

A: Given that,  The functions fn and f are integrable on a,b. fn converges to f on uniformly a,b. an…

Q: For each nN, consider x,, H (a) (b) 71 X with x E R. Define the sequence {gn} by 8n(x) = f(x) = 1…

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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: a) For which values of x does f(x) =E n=0 converge and for which values ofx does it diverge? b) When…

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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: Let pn = 5(6^n) + n^7/6^n. Show that, as n → ∞, {pn} converges to 5 linearly with rate 1/6.

A: Given that pn=5×6n+n76nTo showlimn→∞pn=5and rate of convergence=16

Q: (c) Given f: R → R, suppose that f is differentiable and |f' (x)| < 1 for all x E R. Show that the…

A: This is a question of Real Analysis and Numerical Analysis.

Q: Suppose fn : [0, 1] → R be defined as f,(x) Xn , for n e Z+, for x € [0, 1]. Define: 0, x € [0, 1)…

A: We will be using definition of uniform and pointwise convergence.

Q: 2. s) Let x1 = 1 and define rn+1 = v3+2x, for n E N. Show that the sequence (xn) converges and find…

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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: If f(x) ≤ g(x) for all x in [a, +∞o) and converges. •+∞ f(x) da converges, then [*** g(x) dx also

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Q: Determine whether the series of functions E fn converges uniformly: (-1)n+1 Vn+ cos(x) D = (-0, 00).…

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Q: Compute the point-wise limit of (fn) and determine if they converge uniformly on given intervals. 1.…

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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: E (n!)² (K)! Fan what e is convergent ?

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Q: Let f(x) be a differentiable function such that for all x > 1, f(x) > 1 f(x) and f(x) < lim f(x) = 1…

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Q: (c) Given f: R → R, suppose that f is differentiable and |f' (x)| < 1 for all a E R. Show that the…

A: Please check step 2 for the solution.!

Q: The sequence {G 3n+2 n=1 O None converges to e O diverges converges to Ve? converges to e3

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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: We define an arithmetic sequence of functions: fn: [1,0) → R recursively as: fi(x) = Vx, Vx € [1, 0)…

A: Given fn:[1,∞)→ℝ recursively as: f1x=xfn+1x=x+fnx for all x∈[1,∞). (a). Let x∈[1,∞), therefore:…

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: Prove that if a sequence of continuous functions fn :R→R is uniformly convergent on Q, then it is…

A: Given function fn:ℝ→ℝ sequence of continuous function. fn is uniformly convergent on ℚ To prove: fn…

Q: Let h: [0,1] R be a continuous function with h(1) by fn(x) = x"h(x) for x e [0,1]. Show that {fn}…

A: A sequence of real-valued functions considers two types of convergence. These are pointwise…

Q: The sequence of functions f,: r → na"+1 1. converges uniformly to 0 over [1/2, 1]. 2. converges…

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Q: Use sequence

A: converges.

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

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Q: For n e N, define fn : [-1, 1] → [0, 1] by fn(x) = (1 – x2)". For all 8 e (0,1), define Is = [–1,…

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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: Define Find the values of x where the series converges and show that we get a continuous function on…

A: The series is  . All the   are even and that's why consider .

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: (a) Define Vn E N, an rate constant for which this sequence converges to 0. – 3-" +. Find the order…

A: Given that, ∀n∈ℕ, xn=1n2n-3-n+1nn We have to prove that xn converges to 0.

Q: Let f be a positive, continuous, decreasing function. If /° f (x) dx = 5 , which of the following…

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Q: uniformly co

A: Given, if fn and gn are uniformly convergent sequences of functions. To prove that fn+gn converges…

Q: For any n N, define the function Xn RR by : if x ≥n Xn(2)= 0, if a <n. Show that the sequence of…

A: We will use definition of limit of sequence of function.

Q: Let f(x) = a,x",g(x)=b,x". Suppose both series converge for |x| < R. Let n=0 n=0 x, be a number such…

A: When a series of n terms is said to converge, as the value of n tends to 0, the value of the terms…

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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Q: Suppose that fn(x) and f(x) are functions R –→ R. Suppose also that fn converges uniformly to f as n…

A: Suppose that fnx and fx are functions from ℝ to ℝ such that fn converges uniformly to f as n→∞.  To…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.