1.1 Show that {fn(x)} converges uniformly on [a, b] if and only if given any E > 0 there is a natural number N such that fn(x) - fm(x)| < e for all xE a, b whenever m, n > N.

Q: 3. Define fn on (0, 1) by fn(x) = x". Show that fn converges to the constant function 0, but that…

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Q: Let fn(x) = 1+cos2 2023 (na). Does {f} converge uniformly to a function f on R? √n

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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: Show that it is not true that for every error 1/m there exists n such that |Ek=1 polt – r/(1 - r)| <…

A: We will show that for every m there does not exist any unique n that works for all r in (0,1)

Q: 1. (a) Let an → 2. Prove from first principles (i.e. give a direct e-N proof) that a → 4. (b) Let a?…

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Q: 7. Prove that a converges if kak→ 0 as k→ ∞ for some > 1. Does the result hold if a = 1?

A: The objective of the question is to prove that a sequence converges if the absolute value of its…

Q: b) Suppose (xn) does not converge to 0. Then there exists e > 0 such that all except for finitely…

A: Let us suppose the following sequence xn=1,0,1,0,1,0,..... We have to see that the odd terms are 1…

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: Let fn(x) = sin" x and let 8 > 0. a) Prove: fn converges uniformly on [0, 5 – 8], but not uniformly…

A: The given function is fnx=sinnx. To find: (a) fnx=sinnx uniformly or not on given domains.…

Q: nx 4. Let fn(x) = 1+7²², x € R. Prove that fn → 0 pointwise on R, then check whether fn → 0…

A: Since you have posted a multiple question according to guidelines I will solve first(Q4) question…

Q: Suppose that Z~N(0,1). Then find Pr(2.78 < Z < 2.84)

A: Given data, Z~N(0,1) That is Mean=0 Standard deviation=1 P(2.78 &lt; Z &lt; 2.84)=?

Q: Suppose (xn) does not converge to 0. Then there exists e>0 such that all except for finitely many…

A: Consider the sequence xn=1,0,1,0,1,0,....... Here the odd terms are 1 and the even terms are 0.…

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: Study uniform convergence on 10, 1) of the sequence of functions Jn for the following: a fn(x) =…

A: Solution a: Given that fn(x)=11+(nx+1)3 in 0,1. Now, f(x)=limn→∞fn(x)=limn→∞11+(nx+1)3=0,    x≠012,…

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

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

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.