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Consider the function f(x) = x2 on [-π.π]. a) Find the Fourier Series expansion of f(x) on the given interval. b) By substituting an appropriate value for x, find the sum of the series Σ n=1 (-1)n n² c) By substituting an appropriate value for x, find the sum of the series 00 Σ n=1 1 n² Using this result, what is the sum of just the even terms Σ 1 (2m)2 m=1 d) Use c) to compute 00 Σ m=0 1 (2m + 1)² (What result about infinite series makes this work?)

Consider the function f(x) = x2 on [-π.π]. a) Find the Fourier Series expansion of f(x) on the given interval. b) By substituting an appropriate value for x, find the sum of the series Σ n=1 (-1)n n² c) By substituting an appropriate value for x, find the sum of the series 00 Σ n=1 1 n² Using this result, what is the sum of just the even terms Σ 1 (2m)2 m=1 d) Use c) to compute 00 Σ m=0 1 (2m + 1)² (What result about infinite series makes this work?)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.2: Exponential Functions
Problem 71E
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Consider the function f(x) = x2 on [-π.π]. a) Find the Fourier Series expansion of f(x) on the given interval. b) By substituting an appropriate value for x, find the sum of the series Σ n=1 (-1)n n² c) By substituting an appropriate value for x, find the sum of the series 00 Σ n=1 1 n² Using this result, what is the sum of just the even terms Σ 1 (2m)2 m=1 d) Use c) to compute 00 Σ m=0 1 (2m + 1)² (What result about infinite series makes this work?)
Transcribed Image Text:Consider the function f(x) = x2 on [-π.π]. a) Find the Fourier Series expansion of f(x) on the given interval. b) By substituting an appropriate value for x, find the sum of the series Σ n=1 (-1)n n² c) By substituting an appropriate value for x, find the sum of the series 00 Σ n=1 1 n² Using this result, what is the sum of just the even terms Σ 1 (2m)2 m=1 d) Use c) to compute 00 Σ m=0 1 (2m + 1)² (What result about infinite series makes this work?)
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