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 that the series 1 40+(a, cos(2nz)+b, sin(2xnx)) n-1 converges in L²-metric to f.

Q: Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e²x, a = 0

A: Since you have asked multiple questions in a single request so we will be answering only first…

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Q: Assume that f is continuous and | f) dt 3 sin х — х сos x. 0, (a) Determine f(;n). (b) Find f'(x).

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Q: Ee whether or not fn converges uniformly nclusion. b) (0, 1)

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Q: Theorem 17 (H.w) If lim x, # 0 then E=1Xn diverges series.

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Q: 1. Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e²x, a = 0 b)…

A: Given At  The formula to calculate the Taylor polynomial is:               Now,

Q: Let a not be an integer, and f (x) = cos ax, x E (-7, 7] Extend f to PC (27). Show that 1 2 cot aT =…

A: Using Fourier series and putting some particular value to get our required series.

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Q: e graph of the

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Q: +00 4+ cos n 3n - Vn n=1

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Q: expansion

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Q: b) Determine whether the improper integral 1+ sin x 1+ x2 converges or diverges.

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A: G

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Q: Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e2x, a = 0

A: Since you have asked multiple questions in a single request so we will be answering only first…

Q: 3. Let g(x) = So Expand g as a power series centered at 7. t² sin tdt

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