Let f(x) = (ncosx)/(n+ex) x in [0,1], n in N Prove that the sequence converges uniformly on [0,1].

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Q: * Show that the sequence (xn)n>1 in R, where In = 1- Vn' converges to 1.

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Q: Let f(x) = (ncosx)/(n+ex) x in [0,1], n in N Prove that the sequence converges uniformly on [0,1

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

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Q: Sn(x) = e-(2-n)a

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Q: Let {fn(x)} : „x ER and > 0. 1+ nx Show that ift>0, the sequence {fn(x)} converges uniformly on

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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: Show that the sequence {fn} where fu(r) = e-n* is uniformly convergent on [a, b), a > 0.

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Q: Consider the power series f (a)-Σσ", αε (-1, 1). т € (-1,1). n=0 Show that it converges uniformly on…

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Q: Show that the sequence of functions {f.} where f,(x)=: 1+ nx „xER is non-uniformly convergent on…

A: Show that the sequence of functions fn(x), where  fnx=x1+nx2,  x∈ℝ is non-uniformly convergent on…

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

A: See the attachment

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Q: n En=0m! I)Use Cauchy criterion prove that the sequence does not converge uniformly

A: Given: ∑n=0∞xnn!

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Q: 1. Prove that the sequence converges to . 2n+1

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Q: tan (x) n' In(n+2) 04: Determine the convergence of A=[0,+). on

A: Let the sum of the series be given by Sn. ⇒Sn=∑n=0∞tan-1(x)x2+n2ln (n+2) un=∑ n=0∞tan-1(x)x2+n2ln…

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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: 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: Prove that any convergent sequence {xn}=1 is bounded.

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Q: 2n+1 4. Use the definition to prove that ) converges.

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Q: Suppose {fn}n=1 and {gn}n=1 converges uniformly on E where {fn}n=1 and {gn}n=1 are sequences of…

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Q: 4. Show that the sequence { fn}, where nx fa(x) = xER 1+n°x2 is point-wise convergent but is not…

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Q: Show that the sequence f.(x)3 converges to a limit function (that you should determine) for each xEL…

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Q: If |fn(x)| < an for all x and Ln=1 an converges, prove tha En-1 fn(x) converges uniformly. n=1

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Q: Consider the sequence defined by xo = a > 0 and %3D a X+1 = } ( x* + -) Xk Prove that this sequence…

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Q: Find the pointwise limit f(x) for {nxe-nx} for x ∈ (0, +inf)). Does the sequence converge uniformly…

A: Given, fx=nxe-nx , x∈0,∞ Since, limn→∞fx=0 , ∀x x=0, fx=0 and e-nx→0 as n→∞ Let, fx converges…

Q: The sequence a = is a divergent sequence o converges to 8 none O converges to 0

A: Given: an=8nn For sequence, an, if limn→∞an+1an&gt;1, then sequence is divergent.

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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