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: For each nЄ Z+ let fn : R → R be the function пх fn(2) n|x| +1° (a) Prove that (fn) converges…

A: Given the sequence of functions  defined by .(a)Consider the limit,This is true for every 0≠x∈R.…

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

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Q: re 1. (a) Let fn(x) = (1-x)x". Prove that the sequence converges point-wise on [0,1]. Find the limit…

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Q: Give an example of a function f: -1, 1] → R that is not continuous at 0. Then find an explicit…

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

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Q: 9. Prove that if {fn(x)} converges uniformly to the function f(x) on each of the sets A1, A2, ...,…

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Q: Suppose the function f(x) has a unique zero P in the interval [a,b]. Further, suppose f''(x) exists…

A: Suppose the function f(x) has a unique zero P in the interval [a,b]. Further, suppose f''(x) exists…

Q: Prove that the sequence is monotone and bounded. Then find its limit. s1 = 5 and sn+1 = V4sn +1 for…

A: See the attachment

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: Let ƒ : [−1,1] → R be a continuous function. Prove that the sequence (n) defined inductively by Xn 1…

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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 f: RR be a uniformly continuous function. For n N, define a function fR → R by fn(x) = f (x +…

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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: For each of (a)-(d) give a sequence (rn) with the stated prop- erties. In each case, the sequence…

A: Example and illustrations.

Q: Exercise 2. Determine whether the sequence of functions (fn)n converges uniformly on S. 1 S = [0,…

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Q: 1 Prove: Ex-1 n=1 (n+1)(log(n+1)), converges for p > 1 and diverges for p ≤ 1.

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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.