Consider fn: [0, π] → R given by fn(x) = sin(x), for n ≥ 1. Prove that the sequence (fn) converges pointwise to f: [0, π] → R given by if x = 0 if x # yet this convergence is not uniform. (See Figure 7.3) 1.0 0.8 0.6 0.4 0.2 0.5 1.0 f(x) = 15 2.0 Figure 7.3: The functions f(x) = sin"(x) 2.5 3.0

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: 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: E. i. Suppose that the sn satisfies both limn→∞ s2n = 3 and limn→∞ s2n+1 = 3. (That is, the sequence…

A: i) The sequence sn satisfies both limn→∞s2n=3 and limn→∞s2n+1=3. We have to show that limn→∞sn=3.…

Q: .3 Let fn : [-3,3] → R be defined by fn(x) = . Find the pointwise limit of fn(x). 2n+1' .4 Does…

A: Given,         fn : [-3 ,3] -> R be defined by ,                             fn(x) = x(2n)/{x(2n)…

Q: 2. For n E N, let fn(2) - converges uniformly on [0, 1]? nx +sin(nx?) n 1 (a) Show that the sequence…

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Q: Let fn(x) = exp(-2")+1, for all r E [0, 1], for all n E N. 1. (fa)n converges pointwise to f: [0,…

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Q: . i. Suppose that the sn satisfies both limn→∞ s2n = 3 and limn→∞ s2n+1 = 3. (That is, the sequence…

A: Limit point of a sequence is that point whose neighborhood is surrounded by infinitely many points…

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: 1. shown that I and I norms on R" are equivalent with respect to convergence and show that the…

A: It is given that,  l2 and l∞ are the equivalent norms on ℝn with respect to convergence. We have to…

Q: 65. Suppose that the coefficients of F(x) = anx" are periodic; that is, for some whole number M > 0,…

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Q: 6. Let an = cos. Show that (an) does not converge using the following methods. (a) Using the…

A: Here we are given an=cosπn3 We have to show that {an} does not converge using  (a) the definition of…

Q: 2. Consider the following sequence of closed subintervals of [0, 1]: | 0, 2 1 3 3 and denote the nth…

A: The given function is fn(x)=1  for x in In,0  for x not in In. In denote the nth subinterval. The…

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: So , if z< 0. et (fn)n be a sequence of functions fn : [0, 1] →R defined by fn(r) = nz Then 1. f…

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Q: 3. Find the pointwise limit of the sequence of functions defined on R by the following and determine…

A: To find the pointwise limit of the sequence of functions and determine whether convergence is…

Q: 1.1 Let {fn (2)} = {1 NI +nz ,z ER and 2 > 0. 1.1.1 Find the pointwise limit of the sequence {fn(x)}…

A: To Examine: The series of function Sn(x)=∑k=1∞xk-1 converges uniformly on [-a,a], a∈(0,1) or not.

Q: Question 3 For each nEN, consider xn (a) (b) n == X with x ER+. Define the sequence {gn} by 8n(x) =…

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Q: #2

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Q: Let f = C(R/Z; C), and for n= 0,1,2,... define 2/₁₁ (2) cos(2x)da; by = 2(a) sin(2xnx)da. 0,1] Show…

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Q: 1.1 Let {fn(x)} = { ,x ER and r > 0. %3D + xn 1.1.1 Find the pointwise limit of the sequence {fn(x)}…

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Q: 1.1. For r € (0, 1] (1) = f(r), wher

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Q: Consider the sequence {fn(x)}, where sin(nx) fn(x) = 1+ nx a. Show that fn(x) converges pointwise to…

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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: Show that Show that Show that fexx dx converges for any p = R. In xdx and f′ (x(ln x)(1 + x²)−¹)dx…

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Q: 2) Suppose that f(x) = 5x² over the interval [1, 2]. Let Ax = 1/n. Let x₁=1+iAx. Find the limit of…

A: Given: fx=5-x2.Formula to be used: ∑i=1ni=nn+12 ; ∑i=1ni2=nn+12n+16

Q: Prove by contradiction the scquence {(-1)"+} is divergent.

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Q: For eksample Xn = n² (n+1)! investigate whether the serles Σ 8 n=1 Xn convergent Or divergent. Prove…

A: We will use ratio test to check whether given series is convergent or not.

Q: Q5. Given f(x): (R.) → (R₁) 3x+2 if x 22 5x-1 if x < 2 f(x) = { (a) Let {n} = {2} be a sequence in…

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Q: 5. (a) Suppose 1 and Xn+1 = 2 Show that {n} converges and find the limit. X1 for all nEN.

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Q: Consider the function f(x) defined by the following rule: a. Calculate f(0). b. What if you try to…

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Q: 4. Consider the sequence of functions {fn}=2 where fn: [a, b] R defined for n > 2 by if 0 < x <- fn…

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Q: Question 3. Let g, = nx1/n.2/a] and g = 0. Show that 9 lim Gn. Does the sequence (gn) converge…

A: Let gn=nℵ1n, 2n and g=0. Show that ∫02g ≠limn→∞∫02gn. The characteristic function ℵA is defined as…

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

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Q: by Consider the sequence of functions (fn) defined on the interval [0, 1] given n²x², 0≤x≤ 1/n 1,…

A: As per policy the first three subparts are answered.   The given sequence of functions is…

Q: Let fn(x) = x¹(1+x¹)−¹. Use the uniform norm to determine if the sequence (fn) uniformly converges…

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