FourierSeries-1

Example 1 says that for 0 x 1 let x 2 1 2 1 2 k1 2

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Unformatted text preview: odd F OURIER SERIES ■ 15 19. Example 1 says that, for 0 ≤ x < π, 1 = Let x = π 2 1 2 1 2 + ∞ k=1 + 2 sin((2k − 1)x). (2k − 1)π to obtain 1= ∞ k=1 π 2 sin (2k − 1) (2k − 1)π 2 ∞ π 1 = sin((2k − 1)) 4 k=1 (2k − 1) π =1− 4 1 3 + 1 5 − 1 7 + ··· 1 1 −1.25(−1) + 4(0.25) − √2 + 2(−2.25)(0) + 4(−1.25) √2 1 1 a1 = −1 f (x) cos(πx) dx ≈ 12 + 2(3.5)(1) + 4(3.5) √ + 2(0)(0) + 4(−2.75) − √ − 1.25(−1) 1 1 2 2 √ = 19 1 + 2 24 a2 = = b1 = ≈ 1 −1 f (x) cos(2πx) dx ≈ 3 4 1 −1 1 12 f (x) sin(πx) dx 1 1 −1.25(0) + 4(0.25) − √2 + 2(−2.25)(−1) + 4 (−1.25) − √2 + 2(3.5)(0) + 4(3.5) + 4(−2.75) √ 9+7 2 = 24 b2 = = −1.25(1) + 4(0.25)(0) + 2(−2.25)(−1) + 4(−1.25)(0) 1 12 + 2(3.5)(1) + 4(3.5)(0) + 2(0)(−1) + 4(−2.75)(0) − 1.25(1) 1 −1 1 √ 2 f (x) sin(2x) dx ≈ − 1.25(0) + 4(3.5) 1 12 1 √ 2 + 2(0)(1) + 4(−2.75) 1 √ 2 − 1.25(0) −1.25(0) + 4(0.25)(1) + 2(−2.25)(0) + 4(−1.25)(−1) + 2(3.5)(0) + 4(3.5)(1) + 2(0)(0) + 4(−2.75)(−1) − 1.25(0) 31 12 f (x) ≈ 1 − 12 + 19 24 √ 1 + 2 cos(πx) + √ 9+7 2 24 sin(πx) + 3 4 cos(2πx) + y 4 2 0 -1 -0.5 0 0.5 1 x -2 31 12 sin(2πx) 1 √ 2 + 2(0)(1)...
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