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(a) Consider the following function and its periodic extension: f(x) = sin z], where and F(x) = f(x+nx), Find the Fourier series representation of F(x). Note that L- म where REZ

(a) Consider the following function and its periodic extension: f(x) = sin z], where and F(x) = f(x+nx), Find the Fourier series representation of F(x). Note that L- म where REZ

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.4: Values Of The Trigonometric Functions
Problem 22E
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(a) Consider the following function and its periodic extension: π f(x) = sinx, where M <1 Find the Fourier series representation of F(x). Note that L = and F(x) = f(x+nn). where NEZ 0x0 Σ n=1 (b) Does the Fourier series you found in (a) converge uniformly? If so, prove this using the Weierstrass test for uniform convergence. If not, discuss whether it converges pointwise anywhere in its domain. (c) Use your result in (a) to evaluate the following two series 1 4n² 1 and π 2 n=1 (-1)" 4n² 1 (d) Attempt to differentiate your Fourier series in (a) term by term. Does the resulting series converge uniformly? If so, prove this using the Weierstrass test for uniform convergence. If not, discuss whether it converges pointwise anywhere in its domain. (e) Carefully differentiate the function F(x), as defined in (a), and then find the Fourier series of the resulting function, i.e., find the Fourier series of F'(z). (f) Write 1-2 sentences comparing your series' from (d) and (e), discussing how well the term by term differentiation has worked. (g) Attempt to differentiate your Fourier series in (d) term by term. Does the resulting series converge uniformly? If so, prove this using the Weierstrass test for uniform convergence. If not, discuss whether it converges pointwise anywhere in its domain. (h) Carefully differentiate your answer from (e), and then find the Fourier series of the resulting function, i.e., find the Fourier series of F"(x). You may reuse working from previous questions. (i) Write 1-2 sentences comparing your series from (g) and (h), discussing how well the term by term differentiation has worked.
Transcribed Image Text:(a) Consider the following function and its periodic extension: π f(x) = sinx, where M <1 Find the Fourier series representation of F(x). Note that L = and F(x) = f(x+nn). where NEZ 0x0 Σ n=1 (b) Does the Fourier series you found in (a) converge uniformly? If so, prove this using the Weierstrass test for uniform convergence. If not, discuss whether it converges pointwise anywhere in its domain. (c) Use your result in (a) to evaluate the following two series 1 4n² 1 and π 2 n=1 (-1)" 4n² 1 (d) Attempt to differentiate your Fourier series in (a) term by term. Does the resulting series converge uniformly? If so, prove this using the Weierstrass test for uniform convergence. If not, discuss whether it converges pointwise anywhere in its domain. (e) Carefully differentiate the function F(x), as defined in (a), and then find the Fourier series of the resulting function, i.e., find the Fourier series of F'(z). (f) Write 1-2 sentences comparing your series' from (d) and (e), discussing how well the term by term differentiation has worked. (g) Attempt to differentiate your Fourier series in (d) term by term. Does the resulting series converge uniformly? If so, prove this using the Weierstrass test for uniform convergence. If not, discuss whether it converges pointwise anywhere in its domain. (h) Carefully differentiate your answer from (e), and then find the Fourier series of the resulting function, i.e., find the Fourier series of F"(x). You may reuse working from previous questions. (i) Write 1-2 sentences comparing your series from (g) and (h), discussing how well the term by term differentiation has worked.
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