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…

Q: Suppose f"(x) = (x + 2)°(x + 1)²(x – 2). Which of the following statements about the curv y = f(x)…

A: If f"(x) > 0 , curve is concave up If f"(x) < 0 , curve is concave down So only check the sign…

Q: Consider the function f(x) = 100e* restricted to the domain () ≤ x ≤ 2. a) Extend f(x) to make an…

A: If f(-x)=f(x) then we say f is an even function. Given function is defined in  0≤x≤2 (a) Even…

Q: By using Power series, show that Taurinus's "log spherical" formula…

A: The law of cosines:                      a2=b2+c2-2bccosA where a, b and c are the sides of a…

Q: Let fn(x) on [0, 1]. 1+ nx (a) Does fn converge uniformly on [0, 1]? (b) Check if lim | fn(x) dæ lim…

A:

Q: Let Ln denote the left-endpoint sum using n subintervals. Compute L4 for f (x) = work. 1 a/x-1 on…

A: Given that :- fx=1xx-1 on [2 , 4] L4=?

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

A:

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

A:

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.

Q: e graph of the

A: Here, in part b we have to sketch the graph of the function to which fourier sine series converges…

Q: 1. Expand f(x) = π 0 if 0 < x < 2² ㅠ if <x< ₁ π, 1 if in a half-range: (a) Sine series. (b) Cosine…

A: The function is defined by fx=0if 0&lt;x&lt;π21if π2&lt;x&lt;π We have to evaluate the half range…

Q: Let F(x) = sin (tº) dt for 0 ≤ x ≤ 2. On what intervals if F(x) increasing?

A:

Q: expansion

A:

Q: ii) Obtain the Fourier series of the function g defined by g(x) = | f(t)dt, justifying your method…

A:

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

A: