1. shown that I and I norms on R" are equivalent with respect to convergence and show that the sequence x₁ = (2,2, e-2k cos k) converges to x = (2,0,0) with respect to the 1₂ norm.

Q: (f) Assume that g(x) > bna" is convergent on (-R, R). Instead of assuming that f(rn) = 0 assume that…

A: We will prove the part (f) using two theorems   Theorem (a)Let f(x)=∑anxn be a non-constant power…

Q: 11. If {an} and {b„} are sequences of real numbers, lim,0 an = 5, lim, bn = 8, and fn(x) = a, + bnx,…

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Q: Compute the limits of the sequences sin (n² – 3n + 7) n² +1 | 2n? +п -5 (b) b, (c) c, = n°e (а) а,…

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Q: Consider the sequence {fn(x)}, where sin(nx) fn(x) = 1+ nx Use the uniform norm to show that the…

A: Given: fn(x)=sinnx1+nx To show: fn(x) is uniformly converge on [1, ∞).

Q: 5n 9. Use the L'Hopital's rule to determine if the sequence an = () converges or diverges as limit n…

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Q: 04: A) Consider the sequence (f,(x) of functions f,:R→R defined by: f.x)=Cos"(x) for-a/25xSa/2.…

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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: Write the first five terms of p_j= j/(j+1) and then prove p_j converges to 1. Be sure to explicitly…

A: We have to solve an inequality for apsilon and j.

Q: The seguence of function fn: x→x"+exp (x^) on [o,1] Then [0,1] we have: (fn) on ) neither conuerges…

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Q: shown that 1₂ and lo norms on R" are equivalent with respect to convergence and show that cos k)…

A: Given:  l2 and l∞ on ℝn with respect to convergence are the equivalent norms. To proof:  With…

Q: plz ans both B and C

A: This method is called power method. If modulus of all eigen values less than 1 then the power method…

Q: Prove that the sequence xn has a convergent subsequence. 3n4 sin n – 4n cos n2 + 2 a) Xn 5n4 – 8n –…

A: We have to prove that xn has a convergent subsequence in each of the given cases.

Q: 5a) Show that the sequence (Sn) given by s₁ = 0 and, for n ≥ 1 Sn+1 = √Sn + 2 converges and find its…

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Q: Is {3n/(3n-1)} n=1 to infinity, a convergent sequence in R with the usual metric dR(x,y) = abs(x-y)?

A: The given sequence is .We need to check that the given sequence is convergent or divergent with the…

Q: (a) Prove that (2x + 2) > 1 for all x > 0. (b) Let so > 0, and sn+1= } (2s, + ) for all n > 0. Prove…

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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: 2. Let fn : [1, 2] → R be a sequence of functions defined by fn(x) = x*. Find the pointwise limit of…

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Q: 52. f(æ) = (-1)**(x – 2)". %3D n=1

A: Radius of convergence problem....

Q: · (−1)″ √2n+ 5 n³ +2n +4 n=1 t) Å sin(n) 3 n

A: Given series are: s) ∑n=1∞-1n2n+5n3+2n+4                            t)  ∑n=1∞sinnn3 To prove: Test…

Q: Use the definition of a convergent sequence to show {7-4/n} n=1 to infinity, converges to 7 where…

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Q: Find the radius of convergence and the interval of convergence n=on+3

A: Given

Q: 1.1 Determine whether the sequence {f.(x)} = {; on I = [0, 1]. converges uniformly %3D

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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: 0 s if Let (f, be a sequence of functions f,: 0, 1]-R defined by f.(x)= Then 1515i l 1. converges to…

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Q: Construct a sequence of unbounded functions 9n : [0,1] → R that converges pointwise to a bounded…

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Q: For the function f from part (a), could the Bisection method be used to find a root in the interval…

A: The bisection method requires that the function be continuous on the closed interval [a,b] and the…

Q: 3. Find the interval fconvergence of the Sevies : 1-3(x-4) + 9 x-4)-..

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Q: determine the domain AND the function to converge the serie e converge la serie: E=0 Re(u). n%3D0

A: Consider the given series

Q: 4. Using the formal definition of uniform convergence show that the sequence f(x) = x- converges…

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