N argument. However, do not forget to check the requirements for these theorems! We assume -∞ < a < b < ∞ real and finite and Q(2) rational function with poles at 2ckL,L [a, b]. Before you compute

Lebesgue Dominated Convergence Theorem
Lebesgue Dominated Convergence Theorem

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Lebesgue’s dominated convergence theorem extends the idea of interchanging limits and integrals to lim fn(c)dx = lim fn(r)dz, with a, b ∈ â„ \ {∞} and n ∈ â„• and fn converges point-wise (meaning for each fixed x ∈ (a,b) the sequence {fn(x)} is convergent). Apply this theorem and Theorem 24.1 of Lecture 24. You do not need to prove the theorems nor do an ε – N argument. However, do not forget to check the requirements for these theorems! We assume -∞ < a < b < ∞ real and finite and Q(2) rational function with poles at {2c}kL,L [a, b].
Before you compute the two integrals below, explain why it is f(z) f(z) lim Res Res n→∞ 2z0 0n 2Fz0
for any function holomorphic in punctured neighborhood of z0 and the convergent sequence lim n→∞ an = a with a, an = 2 for all n ∈ â„•. (Hint: think of the contour integral representation of the residue and Lebesgue’s dominated convergence theorem:
Calculate the integral Q(∞) dx with Q(2) = O(2^-2) when 2 = 0 by expressing it as a finite sum of residues. (Hint: consider the limit lim V→0 ∑(n-1)
Calculate the integral ∮ dz with Q(2) = O(z^-1) when 2 ≠0 by expressing it as a finite sum of residues. (Hint: see the hint of part (b).)

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Let (ancos(nx) + bsin(nx)) be the Fourier series of f in the interval [-T,T] in R, and let (2a, cos(nx) + b’nsin(x)) be the Fourier series of the derivative of f. In the proof of the Uniform Convergence Theorem, we show that the Fourier series of f converges absolutely and uniformly. In particular, we show that both series Σ |an| and Σ |bn| converge. We then apply the Weierstrass M-test with Mn = |an| + |bn| to show that the series Σ (an*cos(nx) + bn*sin(nx)) converges. We also show that the series Σ ((an)^2 + (bn)^2) converges by applying Bessel’s Inequality. We use integration by parts and the assumption that f(-T) = f(T) to show that the series Σ (a’n*cos(nx) + b’n*sin(nx)) converges, since both series Σ (a’n)^2 and Σ (b’n)^2 converge for n = 1,2,3,… The sequence {Sn} of the partial sums of the series Σ |an| converges, since Σ |an| is bounded by applying the Schwartz inequality.

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You are watching: N argument. However, do not forget to check the requirements for these theorems! We assume -∞ < a < b < ∞ real and finite and Q(2) rational function with poles at 2ckL,L [a, b]. Before you compute. Info created by THVinhTuy selection and synthesis along with other related topics.

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